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* : Mp --* M again ::::: Ao is a �-valued function and A 1 is a �-valued 1 -form. As usu al we write d = d + dt /\ aja t. Then P = (ijq)(j 2 = dA - iqA /\ A yields, aft er w ritin F 2 = E 1 /\ dt + 82 with rvalued forms E 1 and 82 , the "electric" and "magn etic� parts of the field strength . =
(�I
304
R E L A T I V I T Y , TENSORS, A N D C U RV A T U RE
has the property that
/ (t) = A i t ,
i = 0, . . . 3 ,
It is also clear from the definition of the exponential map that I dy I dt 11 2 = Ai + A� + A� - A5 is a constant along each of the geodesics starting at p and this constant vanishes only for the null geodesics tangent to the light cone, a submanifold of the vector space M� of codimension 1 . By continuity, we conclude that t is a distinguished parameter for each of the geodesics emanating from p. Since the preceding linear equations must satisfy the geodesic equations
j d2 yi i k (y (t» dy dl = r j dt 2 dt dt we must have r� k (A °t, . . . , A 3 t)A j A k = 0 for all t. In particular this holds at p, that is, t = 0, and for all A i . We conclude that
r�k ( p ) = 0
( 1 1 .25 )
at the "pole" of the normal coordinate system. From ( 1 1 . 1 4) we have
Ogij (P) = 0 oy k All first partial derivatives of the metric tensor vanish at the pole !
( 1 1 .26)
As an application of the use of these coordinates, consider the divergence of a vector field v. As in the Riemannian case div v = ( l g l -l/2 ) � [ l g l l /2 V i ]
ox'
At the pole of the normal coordinates we clearly have oloyi l g l ( p ) = 0 and thus at the pole we have div v = 0 V i 10 yi . Consider now the scalar v� ' At the pole v� i = i o vi loyi + v j rij = ovi l oy i . B ut div v and V i i are well-defined scalars, independent of coordinates; we conclude that in any coordinate system ( 1 1 .27)
This in turn means that
and so
which is
a
�k loglg l l /2 ox frequently used formula.
=
r ,ki
( 1 1 .28)
305
HILBERT'S ACTION PRINCIPLE
We h ave already defined the gradient of a function
f to be the contravariant\72 vec i i J j j of = lg aflax = g flli' The Laplacian f is then the scalar f (gI fli ) /i , or, since gl i k = 0, D 2 f -- g ij Iil·l -- gij Iil·l· ( 1 1 . 29) V2 f = g iJ [ a/ax i (af/ax j ) - rfj (aflax k ) ] , or k �] \72 f = gij [ --'!L ( 1 1 .30) ax i ax l rIl ax k an example, consider a surface M2 with local coordinates u I , u 2 , sitting in ]R3 , x2, x3• For each i, i is a function on M. In Problem with Cartesian coordinates i tor (grad!) div grad f =
=
Y
Th u S
_
As
Xl,
X
1 1 .3 ( 1 ) you are to show that
( 1 1 .3 1 )
2 is the surface Laplacian, H is the mean curvature, and N is the unit normal . where Vcular, parti ]R 3 is a minimal surface iff each coordinate function i Theorem (11.32) : M 2
In
C
is a surface harmonic function on
X
M.
11.3c. Hilbert's Variational Approach to General Relativity Although the following approach will work in any dimension, we shall write out ev erything in the case of a 4-dimensional pseudo-Riemannian manifold M4. be the scalar curvature. Since the determinant det Let is negative in a pseudo-Riemannian manifold, the volume form is
R = g ik Rik = g ik R j ijk J(-g)d4x : = J(-g)dx o I\ dx l I\ dx 2 1\ dx 3
g = g iJ
We shall, with Hilbert, take the first variation of the functional
L R J(-g)d4x
for a I -parameter family of metrics. For our purposes, it will be more convenient to
vary
the inverse of the metric
g�J g� " iJ gi " ij =
+ ex
( 1 1 .33)
=
where the dot denotes differentiation with respect to ex at ex
= O. We must compute [:ex J R J(-g)d4x] = J [R J(-g)] "d4x ( 1 1 .34) and where all integrals are over M. Now [R J(-g)] " = [g ik Rik J(-g)] " = [g ik Rid " J(- g ) + R[ J(-g)] " ( 1 1 .35)
306
RELATIVITY, TENSORS, AND CURVATURE
and
[
We then need
g ik R ik ] · = g. ik Ri k + g i k R'ik
Rik . From (9. 1 0), omitting some indices, a rj + r r - r r r jk I, Rik = aax} ax k
( ) ( ) __
I}
__
and so
Rik =
( ��� ) - (���) +
rr + rr - r r - rr
We shall compute everything at the pole of a geodesic normal coordinate system for Since r = 0 at the pole the base metric
gbk •
.
( 1 1 3 6) at the pole. Although erIk) is not a tensor, we claim that is a third-rank tensor. To see this we look at the transformation law (9.4 1 ) for a connection, w' (a) = p - l w (a ) P + P l d P Differentiating and putting a = 0 give w' = p - l w P , and from this the tensorial nature of follows. Thus at the pole we may write
-
( rfk )
.
r
j - r' ij/k j R· ik = r' ik/j
( 1 1 .37)
and since this is a tensor equation it holds everywhere, in every coordinate system. In this equation, all covariant derivatives are with respect to the base metric at a = O . We may then write ( i k r' j ) = d'IV W ( 1 1 .38) = ( i k r' j )
g ik R' ik g ik /j - g ij /k where w r : = g ik r ;k g ir rlj . Look now at the second term in ( 1 1 .35 ) , R[ J (-g ) r To differentiate a determinant we use agjagik = G ik where G ik is the cofactor of the entry gik . This is clear upon expanding g by the kth column. But the inverse matrix satisfies g ik = G ki j g = G ik j g, _
and so
ag ' k _ agik = g' g Likewise ag-I jag ik = gik g-I , that is, agjag ik = - g ik g. Thus a ( - g ) 1/2 = - 1 g g ) 1/2 ( 1 1 .3 9) -2 i k(ag ik and so J( - g ) ]" = ( a (_ g ) I / 2 jag i k ) (ag i k ja a) = -( 1j2 ) gik J(- g )gi k . Thus v(- g )( - �2 gi k V( - g )g i k ( 1 1.40) =
307
HILBERT'S ACTION PRINCIPLE
Finally
o J R vol4 [ !£ da J RJ( -g)d4x ] a=O = J [ Rik - � g i k R ] gi k J ( _ g)d4X + J divWJ(-g)d4x By choosing a variation gik that vanishes outside some compact subregion of M and =
( 1 1 .4 1 )
applying the divergence theorem to a slightly larger region, we see that the last integral vanishes. Thus
o 1Mr R vol4 [!£ da 1Mr R J(-g)d4X] a=O L [ Rik - �gik R] gik J( _ g)d4X =
=
( l l A2)
for all variations with compact support. We define the (Hilbert) action for the gravitational field by Sgrav =
L Lgravd4X
: = (8JT K ) - 1
L R vol4
( 1 1 .43)
(where K is again the gravitational constant), a nonlinear functional of the metric tensor. Let Snongrav be the action for the nongravitational fields that might be present, such as the electromagnetic fields; it is given by some Lagrange density Snongrav
Ld4x
= r 1M
=
r ff vol 4 1M
where L = cs:>y'( The variational or functional derivative J of the metric is defined through S= M
Ld4x
-g).
oS
=
0
oS /og ik of a functional
r1M Ld4x 1Mr ( oglOLk ) g ikd4x =
( 1 1 .44)
where the variation is assumed to have compact support. In other words, putting aj and so forth,
/ax j ,
oLik = rJ Lik - [ rJ Lik ] + [ rJ Li ] og rJ g rJ (g ) rJ (g k ) the usual Euler-Lagrange expression. Thus, from o ik [R;k - "21 gik R] ( -g) og
f,j
:=
.
,}
is
L g rav
= (8JT K )
'J
,}r
'jr
( 1 1 .4 1 )
� Y
_I
( 1 1 .45)
The (stress)-energy-momentum tensor of the gravitational field is defined to be o (since gravitation is a fictitious field) ; that of the nongravitational fields, 1ib is de fined by
\ - g ) _ o og ik
� _ ) :=
T,. . Ik Y
L nongrav
( 1 1 .46)
308
RELATIVITY, TENSORS, AND CURVATURE
The total Lagrangian i s L = L grav + L nongrav , and so
:g�k {
[ � ] - 1jk}
= (8Jr K ) - 1 Ri k - g i k R
J ( -g)
Then Einstein's equations are equivalent to Hilbert's action principle
8
L [(8Jr K ) - 1 R + .S:']voI4
=
0
(1 1 .47)
It is natural to call R .J ( -g) the Lagrangian of the gravitational field. To understand the geometric meaning of Einstein's equations we must return to OUf study of second fundamental forms and curvature. We proceed to these matters in OUf next two sections.
Problems
1 1 .3(1 ) Use Gauss's surface equations to prove (1 1 .3 1 ) .
1 1 .3(2) (i) Let v b e a vector field i n IR 3 defined along a surface M 2 in IR 3 . I f x 1 , x 2 , x 3 , are cartesian coordinates for IR3 , we define the vector integral 11M vd S to be the vector w with components w i = 11M vi dS. Show that if M2 is a closed surface with unit normal N, then
fL
H Nd 8 = O
We considered the special i ntegral JJ Nd 8 = II dS directly before Euler's equation (4.45). For a closed su rface M we have
11M
Nd 8 = O
since, for example, 11M N 1 d8 = .rfM dy /\ dz = O. Thus, for any closed surface i n IR 3 , not only is the su rface average of N zero, which is geometrically "clear;' but also this average, when weighted by the mean curvature, also vanishes! 1 1 .3(3) Let !.e m : =
(-g) - 877: Fij F IJ. . V� 1
define the Lagrangian for the pure electromagnetic field, with associated action
8em
:=
r F;j Fr sgri gSj J(-g ) d4X _� 8 77: JM
Show (recall i ng that F;j is independent of the metric) that the stress-energy momentum tensor for the electromagnetic field is M i n kowski's
(Recall that locally F2 = dA1 , where A is the covector potential; we shall se e later on that A is usually globally defined. Thus 8e m can be expressed as a functional of A and the metric. We shall see in Section 20.2c that 8 8em/8 A == 0 is simply a statement of Maxwell's equations in free space. Thus one obta in s
309
THE SECOND FUNDAMENTAL FORM IN THE RIEMANNIAN CASE
the equations of motion of the electromagnetic field, Maxwell's equations, by putting the fi rst variation of the total action with respect to the potential equal to 0 13
� [ Sgrav + Seml = 0 J
Thus one varies the metric potentials in the total Lagrangian to obtain the gravi tational (Einstein) field equations and one varies the electromagnetic potentials to obtain the electromagnetic (Maxwell) field equations!)
1 1 .4. The Second Fundamental Form in the Riemannian Case
If you fold a sheet of paper once, why is the crease a straight line?
11.4a. The Induced Connection and the Second Fundamental Form Let V ' C Mil be a submanifold of a Riemannian manifold M . If we restrict the Rieman nian metric of M , ( , ) , to vectors tangent to V , we obtain a Riemannian metric for V , the induced metric. Let V be the Riemannian connection for Mil and let v n - I be an (n - I ) -dimensional hypersurj'ace of M . Define a new connection for V as follows. Let X be tangent to V at " p and let Y be a vector field tangent to V near p . Let x I , . . . , x and u I , . . . , U"- I , be local coordinates for M and V , respectively, near p . Then V x Y = X a VY / au a makes
N
__��------ir--
______
M
y
v
Figure 1 1 .3 sense since Y is a vector field defined along V . Let N be a unit vector field along V " - 1 that is normal to V and let Z be any vector field defined along V (it needn' t be tangent to V). Define, at p in V V Vx Z : = projection of VxZ into the tangent space V"
( 1 1 .48)
= VxZ - ( V x Z , N ) N .
In particular, to the vector fields X and Y tangent to V we associate another tangent vec tor field V v x Y. One checks immediately that (9.2) is satisfied by V v and thus ( 1 1 .48) define s a connection for V " - I . We claim more: V V is the Riemannian connection for the induced metric on V " - I . You are asked to prove this in Problem I l .4( 1 ) . Notice that we have merely imitated Levi-Civita's construction in the case of a surface V 2 C ]R 3 .
310
RELATIVITY, TENSORS, AND CURVATURE
What i s the generalization of the second fundamental form? We proceed a s i n Section 8. l b. In the following X, Y, Z, are tangent to Vn- I and N is a local unit normal to V. We define by
b(X) Put
B (X, Y)
= :=
-VxN
( 1 l .49 )
( X, b ey) )
Extending X and Y to be fields on M , we have
B (X, Y)
=
(X, b eY ) ) = (X, - V y N) = Y « (X, -N) ) - (V y X, - N) = (Vy X, N) = ( VxY, N) = B (Y, X)
( 1 1 .5 0 )
(why?). Then b is again a self-adjoint linear transformation; b has (n - 1 ) real eigenval ues K I , , KI - 1 called principal (normal) curvatures. The eigen directions are called l the principal directions, and they can always be chosen to be mutually orthogonal. From ( 1 1 .48) and ( 1 1 .50) we have the Gauss equations •
•
.
VxY
=
V � Y + B (X, Y)N
( 1 1 .5 1 )
generalizing the surface equations (8.30). We shall say that vr C Riemannian Mn is geodesic at p provided every M -geodesic through p, tangent to V at p, lies wholly in V . Thus all of the -geodesics through p are also M-geodesics !
V
p
. . ..:::: . .
Mn
.
r plane in
M;
p
Figure 1 1 .4 Then V (if connected) is made up of geodesic segments of M emanating from p, tangent to an r -plane in M; . A plane in JR. 3 and an equatorial r-sphere s r in sn are examples. Unlike in these examples, it is not true in general that a V -geodesic sta rting
at a point differentfrom p will still be an M-geodesic. If V"- I is geodesic at p, then at p B (X, X)
=
( VxX, N)
=
0
( 1 1 .52)
since X can be extended to be the tangent to a geodesic of V that is then also a geodesi c of M . Thus the secondfundamentalform B of V at p is identically 0 if V is geodesic at p.
311
THE SECOND FUNDAMENTAL FORM IN THE RIEMANNIAN CASE
2 As in the case o f a V
c
lR\ w e define the mean curvature H o f V " - 1 H
:=
tr b = K I + . . . + KI1 - 1
c
M il by
and thi s is again significant for considering variations of the
(n - 1 )-volume of V " - 1 • (In fact, you should be able to guess the generalization of Gauss's formula (8.26).) V n - 1 C M n is said to b e a minimal submanifold o f M if H vanishes at all points o f V . V i s geodesic a t every point p o f V (we then say that V i s totally geodesic) N o te th at if
then V is a minimal submanifold of M . Thus the equatorial S" - 1 C S" is minimal in S" . (Note, however, that S 2 does not have minimum area in S 3 ! ) The other invariants L"'
lR 3 •
•
.
.
not called the Gauss curvature. We shall talk more about some of these matters in our
next section on relativity.
1 l.4b. The Equations of Gauss and Codazzi Mil has a connection V and curvature tensor sor R v re
tu
ten
R V (X, Y)
:=
R;
V " - 1 has a connection V V and curva
[V � , V � J - V �. Y l
are their curvatures related? In other words, if X, Y, and Z are tangent to V n - I , how are the vectors R (X, Y)Z and R V (X, Y)Z related? RV (X, Y)Z is certainly tangent to V but there is no reason why R (X, Y)Z should e b . We can see their relation as follows. " Let a" = a/ a u , ex = 1 , . . . , n - 1 , be a local coordinate basis for v n - I . Since these fields can be considered as vector fields defined along the sub manifold V, we have, from ( 1 0.2) How
On the other hand, [V �. V �� - V �� v �J ay =
R V ( a""
a tl ) Oy
insert va. a y = v �" a y + ( Vau a y , N) N , and take second derivatives, using = - b ( 0tl ) . By a calculation entirely similar in spirit to Gauss 's and yours in Problem 8.5( 1 ) we get Now
VOp N
[ Va. V a� - V a� Vau J Oy =
[ V �" V �# - V �# v �J ay
+ B (a""
The expression i n the curly braces
a y ) b ( 0tl ) - B ( atl , oy ) b ( a,, )
{ } can b e simplified. Our prescription ( 1 1 . 1 3) for covariant derivative of a covariant tensor field can be shown to be equivalent to the following version of Leibniz 's rule. For any p-times covariant tensor T, for vector taking the
312
R E L A T I V I TY , TENSORS, A N D C U R V A T U R E
X , and for vector fields Y 1 , . . , Y p ' then T (Y 1 , . . . differentiate it with respect to X . Then ( 1 1 . 1 3 ) says .
XT (Y 1 , · · · , Y,,)
=
(VX T) (Y 1 ,
•
•
•
, Y ,, ) i s a scalar field and w e may , Yp)
(1 1.54)
(with a similar rule for any mixed tensor) . Apply this to the manifold VIZ- l and the covariant tensor B to get
a -a B (8 (3 , 8 y ) au
=
(Vav" B ) (8(3 , 8 y ) + B (V �" 8f! ' 8 y ) + B (8fJ , V�" 8 y )
Thus the expression i n braces { } in ( 1 1 .5 3 ) becomes, using
(10.1),
(11.55)
(V �" B ) (8(3 , 8y ) - (V �� B ) (8a , 8 y ) = BfJ y//a - Ba Y //(3
where we use the double slash II for covariant differentiation using the connection V V . (This should b e n o surprise after Problem 1 1 .23.) Then ( 1 1 . 5 3 ) can b e written
R (8a , 8(3 ) 8 y
=
R V (8a , 8(3 ) 8 y + B (8 a , 8 y)b (8(3 ) - B (8 (3 , 8y)b(8a )
(11.5 6)
+ [ BfJ y// a - Ba Y // (3 ]N a Finally, we may multiply by x y fJ z y and sum over ex , {3 , and y to get
(11.57)
R (X, Y) Z = R V (X, Y) Z + B( X, Z )b(Y) - B (Y, Z )b(X) + [(V� B ) (Y, Z ) - (V� B ) (X, Z ) ] N
which is a Riemannian generalization of ( 8 . 34) . On the right-hand side, only the last line is a vector normal to V . Since X, are tangent to V , we have two consequences. First
( R (X, Y)Y, X)
=
Y , and Z
( R v (X, Y)Y, X) + B (X, Y) (b(Y) , X) - B (Y, Y) (b(X) , X)
or
(11.5 8)
( R (X, Y)Y, X) = ( R v (X, Y)Y, X) + [B( X, y) ]2 - B (Y, Y) B (X, X) Now note that if we make a substitution, X then it is easy to see that
f--+
X'
=
a X + bY and Y �
Y'
=
cX + dY ,
( R (X', Y')Y', X') = (a d - bc)2 ( R (X, Y)Y, X) On the other hand, if we let II X /\ Y II denote the area of the parallelogram spanned by
X and Y
II
X /\ Y 1 1 2
=
II
X 11 2 11 Y 1 1 2 sin 2 L X, Y
= I I X 1 1 2 11 Y 1 1 2 - (X, y) 2
THE SECOND FUNDAMENTAL FORM IN THE RIEMANNIAN CASE
then under the substitution we have
II X'
1\
Y'
11 2 = (ad - bc ) 2 II
X 1\ Y
313
11 2 . Conse
if X and Y are independent and if we let X 1\ Y denote symbolically the 2-plane spanned by X and Y, we then have that
qu entl y,
( 1 1 .59) K (X 1\ Y) (R(X, Y)Y, X) II X 1\ Y 11 - 2 depends onl y on the plane X 1\ Y and not the basis X, Y itself. This number, which is a :=
function of 2-planes in the tangent spaces to Mil , is called the (Riemannian) sectional cu rvature for the plane X 1\ Y. By taking X and Y to be orthonormal, ( 1 1 .58) can be wri tten
KM (X 1\ Y) = Kv (X 1\ Y) + [B(X, y)] 2 - B(Y, Y) B (X, X)
( 1 1 .60)
1 w hich we shall call Gauss's equation for the hypersurface V"- C Mn . Our second consequence of ( 1 1 .57) is what we shall call the Codazzi equation
(R(X, Y)Z, N)
=
(Vi B)(Y, Z) - (V� B) (X, Z)
( 1 1 .6 1 )
We now will show that these two equations reduce to the surface equations of the same name.
H.4c. The Interpretation of the Sectional Curvature S uppose now that we consider a submanifold vr C Mil that need not be of codimension 1. We define, for any vector field Z defined along V and for any vector X tangent to V
at p
viz : = projection of VxZ into the tangent space V;
The induced connection for vr is again defined at p E V by applying this formula in the case that Z = Y is tangent to V . The normal space to vr at p, ( V,, ) -1 C M" now has dimension n - r ; let NA, A = 1, . . . , n r , b e normal vector fields along V that are orthonormal. These will exist in some small V -neighborhood of p. Then -
viY := VxY - L(VxY, NA ) NA A
( 1 1 .62)
For each normal NA we shall define a second fundamental linear transformation bA Vp -+ V" by
:
( 1 1 .63) (Note that although VXNA is orthogonal to Nil , we need ViN A in order to assure that it is tangent to V ! ) A calculation similar to that leading to ( 1 1 .60) will now lead to Gauss's equations
KM (X 1\ Y)
=
Kv (X 1\ Y) + L{[BA (X, y) ] 2 - BA (Y' Y ) BII (X, X) } 11
( 1 1 .64)
Now let X, Y be any orthonormal pair of vectors tangent to Mn at a point p. Consider C Mil generated by all the geodesics of M that are tangent to the 2-plane X 1\ Y at p. This surface is geodesic at p, and just as in ( 1 1 .52),
the 2-dimensional surface V 2
314
RELATIVITY, TENSORS, AND CURVATURE
all second fundamental forms must vanish at p, b A = 0 at p. Thus, from ( 1 1 . 60 ) KM (X /\ Y) = K v (X /\ V) . Putting X = e l , Y = e 2 , we see K M (X /\ Y)
= =
K v (X /\ Y)
( R v ( e l ' e2 ) e 2 , e l )
=
R i2 l 2 = K
is the Gaussian curvature of V 2 with its induced Riemannian metric. Thus
is the Gaussian curvature of the 2-dimensional geodesic disc V 2 generated by the geodesics of Mn that are tangent to the plane
Theorem (1 1.65) : KM (X /\ Y) X /\ Y.
In the special case when M 3 = lR 3 and V 2 is a surface in lR 3 , K v (X, Y) is simply the Gauss curvature K v = R V 1 2 1 2 of V 2 and ( 1 1 .60) says that 0 = K + ( b I 2 ) 2 - bl l b22 = K det b, since X and Y are orthonormal. This is Gauss's theorema egregium. For the Codazzi equations ( 1 1 .6 1 ), in our V 2 C lR 3 case, R = 0 and the right-hand side say, from ( 1 1 .55), bpy//a = bayllfi . From Problem 1 1 .2(3) this is the usual Codazzi equation. -
1 1.4d. Fixed Points of Isometries Let ct> : M n ---+ Mil be an isometry. The fixed set, that is, the set F = { x E M I ct> (x ) = x } of points left fixed by ct> , can consist perhaps of several connected pieces or "compo nents." Consider two points x and y in F and consider the minimal geodesic y joining x to y . We know from ( l 0.25) that such a minimal geodesic will exist if x and y are sufficiently close, and furthermore this minimal geodesic is unique, again if x and y are sufficiently close. S ince the length of ct> ( y ) is the same as the length of y , we see that ct> (y ) is again a minimal geodesic joining x to y . By uniqueness ct> (y ) = y , that is, the entire minimal geodesic joining x to y lies in the fixed set F provided that x and y are in F and sufficiently close. In other words, if two fixed points of an isometry are sufficiently close, then the entire geodesic joining them is fixed. It is not difficult to see then (see [ K ] ) that in fact the fixed set of an isometry consists of connected components, each of which is a totally geodesic submanifold.
As an example, the isometry of the unit sphere x 2 + y 2 + Z 2 = 1 that sends (x , y, z) to (x , y , - z) has the equator as fixed set. The "same" isometry of lR P 2 has fixed set consisting of the "equator" and the "north pole."
Problems
1 1 .4(1 ) Let X, Y, Z, be tangent vector fields to vn -1 . Extend them in any way you wish to be vector fields on Mn . Show that (i) V�Y - v�X is the Lie bracket [ X, Y] on V and thus the connection V V is symmetric.
315
THE GEOMETRY OF EINSTEIN'S EQUATIONS
(ii) Show that X (V, Z) = ('��V, Z)
+ (V, V�Z)
and hence V v is the Levi-Civita connection for
V.
1 1 .4(2) If you fold a sheet of paper once, why is the crease a straight line?
1 1 .5. The Geometry of Einstein's Equations
What does the second fundamental form have to do with the expansion of the universe?
11.5a. The Einstein Tensor in a (Pseudo-)Riemannian Space-Time Let eo , . . . , e3 be an "orthonormal" frame at a point of a pseudo-Riemannian M4. The following relations can be found in [Fr, chap. 4] ) . There are sign differences from the Riemannian case (considered in every book on Riemannian geometry). Recall that a null vector X has (X, X) = O. For any nonnull vector X we define its indicator E (X) = sign (X, X) . If ei is a basis vector we shall write E (i ) rather than dei ) ; thus E (O) = - 1 . The Ricci tensor in its covariant form defines a symmetric bilinear form i Ric(X, Y) : = Rij X Y j
In particular
( 1 1 .66)
The Ricci quadratic form can be expressed in terms of sectional curvatures Ric (ei , ei )
=
E (i ) L K ( ei Hi
1\
ej )
( 1 1 .67)
th at is, the Ricci curvature for the unit vector ei is (except for a sign) the sum of the sectional curvatures for the (n - I )-basis 2-planes that include ei . In particular, for a Riemannian swiace M l , Ric (e l , e , ) = K (el 1\ el) = K is simply the Gauss curvature. The scalar curvature R is also the sum of sectional curvatures R
=
i Ri =
L
i .j. with i =J j
In the case of a surface R = K ( e l 1\ el) + K (el The Einstein tensor is defined to be
K (ei 1\
el )
1\
ej )
=
( l 1 .68)
2K. ( 1 1 .69)
with associated quadratic form G (X, X) = Rij Xi X j - ( l j2) (X, X) R . One then has th at the Einstein quadratic form is again a "sum" of sectional curvatures, G (ei , ei ) =
316
RELATIVITY, TENSORS, AND CURVATURE
-E (i ) L K (ef) , where ef i s a basis 2-plane that i s orthogonal to e; . For example, for the timelike eo G (eo, eo) = K (el /\ e2 ) + K (e l /\ e 3 ) + K (e2 /\ e 3 ) 871:K T (eo , eo) = K (e l /\ e2 ) + K (el /\ e3 ) + K (e2 /\ e3 ) The second equation follows from Einstein's equation ( 1 1 .9).
( 1 1 .70)
In particular, if we are dealing with an electromagnetic field, the energy-momentu m tensor (as given in Problem 1 1 .3(3» is
( 1 1 .7 1) Let us write out Too = T (eo , eo) in the case of Minkowski space. (We continue to use the convention that Greek indices run from I to 3 while the Roman run from 0 to 3 ; unfortunately this is counter to the notation in most physics books.) First, from Equation (7. 1 8), note that FOk Fo k = FoOl FO Ol = FOOig Olf3 FOf3 = EOI E OI = E 2 . A lso Fr s F rs = 2(Fop F 0f3 + L OI
( 1 1 .72) Thus in Minkowski space, Too = (471: ) - 1 [E 2 + ( 1 /4)2(B 2 - E 2 ) ] , or
Too
=
1 871:
- (E2 + B
2
)
( 1 1 .73)
which is the classical energy density of the electromagnetic field (see Problem 1 1 .5(1» . In general, Too is called the energy density of the nongravitational fields, as measured in the frame e, and will be denoted by p
( 1 1 .74) Einstein's equation ( 1 1 .9) implies that the indicated sum of sectional curvatures is a
measure of the total nongravitational energy density ! 1 l .Sb.
The Relativistic Meaning of Gauss's Equation
In the space-time manifold M 4 we may introduce local coordinates x O = t , X I , x2, and x 3 in many ways. After such a selection has been made, the submanifolds y3( t ) defined by putting x O = the constant value t are called the spatjal slices of the coordinate system. These spatial slices are spatial in the sense that (X, X) > 0 for each nonzero tangent vector to Y et ) . On the other hand, the "unit" normal N to Y (t) will always be a timelike vector, (N, N) = - 1 . Of course we could also consider other hypersurfaces, such as, those where X l = constant and N is then spacelike, but our main concern here is with the spatial slices. The reader may refer to chapter 4 of [Fr] for further discussion. Let N = eo be the unit normal to the spatial slice Y\t). Complete N to an or thonormal basis. We may consider the second fundamental form b of Y (t) , defined as in Section 1 1 .4. We must now, however, be very careful with "signs." For example,
THE GEOMETRY OF EINSTEIN'S EQUATIONS
'f e i s
31 7
an
"orthonormal" basis, then when we expand a vector in terms of this basis, � ei v i , we get v a = (v, ea ) but v a = - (v, eo ) ! Thus for our spatial slice V (t) we have, rather than ( 1 1 .48) ,
� ::::
VxY = V�Y - (VxY, N)N = V�Y - B (X, Y)N
( 1 1 .75)
This will then introduce minus signs into the Gauss equation ( 1 1 . 60)
KM (X 1\ Y)
=
Kv (X 1\ Y) - [B(X, y)] 2 + B(Y, Y) B (X, X)
( 1 1 .76)
We must now make a comment about self-adjoint linear transformations, for ex ample, b, in the case of our pseudo Riemannian metric ( , ) . When M is pseudo Rieman nian, the proof in Problem 8.2( 1 ) of the fundamental theorem on self-adjoint transformations A : JR.n -+ JR.n fails because the scalar product is not positive definite. The cruci al point is that in this case the "unit sphere" (x, x) = I is really a hyper boloid, an d is thus not compact; there is, e.g., no assurance that the continuous function j(x) == (x, Ax) will attain its maximum at any point of this hyperboloid ! Thus a self adjoint A need not have real eigenvalues ! For example, in Minkowski 2-space with metric diag( - 1 , 1 ) the linear transformation with matrix -
is self-adjoint (since its covariant version is symmetric) with eigenvalues ± i . We, however, are concerned here with the self-adjoint b that maps the tangent space to V (t) into itself. Since V (t ) is spacelike, V (t) is a Riemannian submanifold of the pseudo Riemannian space-time, and thus b will have 3 real eigenvalues, and the corresponding eigenvectors, the principal directions, can be chosen orthonormal. By applying ( 1 l .76) to an orthonormal basis of eigenvectors e l , e2 , and e3 of b, we get
( 1 1 .77) Put this now into ( 1 1 .74), where the sectional curvatures K there are for K = KM • Einstein's equation becomes 8 7r KP =
or, from ( 1 l .68)
+ Kv ( e l 1\ e3 ) + K V (e2 + (K I K2 + K I K3 + K2 K3 )
K V (e l
1\ e 2 )
M 4 , that is,
1\ e ) 3
( 1 1 .78) We shall think of this as the geometric version of Einstein's equation involving Too . Let us put it in the proper perspective. For a Riemannian surface V 2 C JR. 3 we have K == K I K2 , which we may now write as
I
0 = - R v - K I K2 2 This is simply Gauss's theorema egregium, and, as we have just seen, is a consequence of the fact that the Einstein tensor G of the flat JR. 3 vanishes.
318
RELATIVITY, TENSORS, AND CURVATURE
Consider a 3-dimensional submanifold V 3 of the flat euclidean 4-space ]R4. The statement that the Einstein tensor G of ]R4 vanishes can be written
0 1 .79) This is a 3-dimensional version of Gauss's theorema egregium . If we consider instead a 3-dimensional space/ike submanifold V 3 ski space, then there is only a simple sign change, yielding I 0 = 2 Rv
+
(KI K2
C
Mri of Minkow_
+ K ] K3 + K2 K3 )
This is the theorema egregium for such a hypersurface of Minkowski space. Consider now a 3-dimensional spatial section V 3 in the actual space-time manifold M4 of our physical world. Einstein's equation 0 1 .78) then says that the combination ( 112) R v + (K] K2 + K I K3 + K2 K3 ) is not 0, as it was in Minkowski space, but is rather a measure of the total nongravitational energy density of space-time!
Note that R v is an intrinsic measure of curvature of the spatial section V 3 , since it is constructed from the Riemann tensor of the Riemannian V 3 . On the other hand, the Ka 's, being principal normal curvatures, measure how V 3 curves in the enveloping M4; thus (KI K2 + K I K3 + K2 K 3 ) is a measure of extrinsic curvature. As J. A . Wheeler put it, Einstein's equation ( 1 l .78) may be stated as follows: The sum of the intrinsic and the extrinsic curvatures of a spatial section is a measure of the nongravitational energy density of space-time.
Finally, I wish to elaborate on ( 1 1 .78), putting it in the spirit of Gauss's theorema egregium. Let p be a point of space-time and let N be a given unit timel ike vector at p . Let V 3 be any spacelike hypersurface that is orthogonal to N at p ; only its tangent plane at p is prescribed. V 3 will have a scalar curvature R v at p that depends strongly on the choice of V 3 . V 3 will also have normal principal curvatures Ka at p, and these again will depend on the choice of V 3 . Gauss's generalized theorema egregium states that the combination ( 1 j2) R v + (K I K2 + K I K3 + K2K3 ) does not depend on the choice of V 3 , but is in fact equal to the value G (N, N) = R ij N i N i + ( l j2) R of the Einstein quadratic form for M4 evaluated on the given normal !
H.Sc. The Second Fundamental Form of a Spatial Slice Consider in space-time M4 a coordinate system in which the metric assumes the fonn ( 1 1 .80)
Thus gOfJ = 0 and ga{3 = h a{3 is the Riemannian metric induced on the slice V 3 (t) defined by putting t constant. (Such coordinates always exist; e.g. , if we take an i ni ti al
319
THE GEOMETRY OF EINSTEIN'S EQUATIONS
sl ice V 3 (0) _ - l !) As we proceed along the t-lines we may contemplate goo
and introduce Gaussian geodesic coordinates as in 1 0.3a, we can even make
Pu t then N
cp : = ==
1 /2 ( goo )
a/atga{3 (t , x) .
cp- I Ot is the unit normal to each spatial slice V 3 (t) . a (h a{3 ) / V (cpN) 0 / 0 V (cpN) a, ' a fJ ox fJ at \ ox \
)+
=
)
Thus
(1 1 .8 1 ) In words, bafJ is essentially the measure of the rate of change of the spatial metric h afJ as one
moves along the normal to the slices, that is, in time!
I t should be clear from this that the second fundamental form will play a crucial role
in discussing the expansion a/ the universe (see [Fr, chap. 1 2] ) . Equation 0 1 .8 1 ) is useful i n the Riemannian V"- 1 C Mn case a s well. See Problem
1 1 .5(2).
1 l .Sd.
The Codazzi Equations
S o far, in this section, we have discussed mainly the geometry of the Einstein equation
Goo
= 8JTK 7(1O, where Too is the (nongravitational) energy density. We now wish to discuss the geometry of GOfJ = 8JT K 7(1fJ . Recall that we have already demonstrated certain symmetries of the covariant Rie mann tensor; for example, R ijkl is skew in (ij ) and also in (kl). The latter is Equation (9 .54). Using the Bianchi identity, you are asked in Problem 1 1 .5(3) to show that there is also the symmetry 0 1 . 82) Back to relativity. Assume a metric of the form ( 1 1 . 80). The Codazzi equations are given in ( 1 1 .6 1 ). If you write these out in coordinate form (as you are asked to in Problem 1 1 .5(3» you will get 1 -b ( -goo ) - / 2 ROy afJ - b yfJ//a y a // fJ
0 1 .83)
320
RELATIVITY, TENSORS, AND CURVATURE
the double slash again denoting covariant derivatives i n V 3 ( t ) (recall that b i s a ten sor on V 3 (t) , not M4). Then
b llf3 //a - b lla // p
=
hIl Y (b yf3 // a - b y a // f3 )
=
h ll y - I ROY af3
where (hIl Y ) is the inverse matrix to the 3-dimensional tensor (h aP ) ' Since (g ij ) i s a matrix of the form
we may write
A. - l h Il Y ROy ap - A.'f' - I g Il Y ROyap - 'f'A. - I g ll i ROi af3 - -'f'A.- I R Il Oaf3
'f'
and so RIl Oaf3
Since goP
=
- (bllp //a - bll a // f3 ) . Then ROf3 = R i OiP = R a Oaf3 = _ (b a f3 //a - ba a // p ) 0, Einstein ' s G Of3 = 8nK Top gives 8n K Tof3 = V (-goo) (Ho"'p - ba f3 ) //a =
( 1 1 .8 4)
which perhaps should be called the Einstein-Codazzi equation. In the case of electromagnetism, in Minkowski space, Top = (8n) - 1 FOk F/ , and FOk F/ = FOa Ff3 a = - Ea Ff3 a = _ E a Fpa = E a Faf3 = E a Baf3 = the f3 th component of iE�g 2 , that is, E x B. By Problem 1 1 .5( 1 ), this is the negative of the momentum density of the field. In general, - TOf3 is defined to be the f3 th component of the momen· tum density of the nongravitational fields and the Einstein-Codazzi equation ( 1 1 .84) relates this to the second fundamental form of the spatial slice. -
H.Se. Some Remarks on the Schwarzschild Solution We refer the reader to [Fr, chap. 5] for details of the following. The Schwarzschild solution is a static solution of Einstein 's equations corresponding to the gravitational field exterior to a single spherically symmetric static mass ball ( e .g the region outside the sun) in an otherwise empty universe. It is not hard to see that the metric for the entire universe must be of the form .,
( 1 1 .85)
in spherical coordinates r, e, with the mass center at the origin. Note that dr does not measure radial distance from the origin; the unknown yrg6 rr d r doe s ! On the other hand, r 2 (de 2 + s in 2 e d 2 ) is exactly the standard metric on the 2-sphere S 2 (r) of radius 3 r in JR. (i.e., the sphere of constant Gauss curvature K = l / r 2 ). This sphere has area 4n r 2 . Thus r is a radial coordinate that is normalized not so that it is distance from the origin but rather so that the 2-sphere r=a has area 4na 2 . The metric coefficient g rr can be obtained as follows. From ( 1 1 .78) we see that R v + 2(KI K2 + K I K3 + K2 K3) = 1 6nKp, where V 3 is the spatial slice t = constant and R v is the Ricci scalar curvature of V . But, from ( 1 1 .8 1 ), the second fundamental fonn
321
THE GEOMETRY OF EINSTEIN'S EQUATIONS
f a sp ati al slice vanishes in a static universe. We conclude that
oarti cul ar
Rv
=
RI!
0 in the region outside the ball of matter.
=
1 67rKp and in
P We wish then to determine the metric coefficient g rr on the spherically symmetric 3 = 1 67r Kp (r) . We may try to realize such a Riemannian V 3 as an embedded V with R v
(again called V 3 ) in euclidean IR,4 = IR, X IR, 3 with coordinates w, r, e , ¢ , which respects spherical symmetry, that is, i s invariant under the rotation group SO(3) IR,3 . ac ting on the space 3_ manifold
w
Figure 1 1 .5 We assume a graph of the form w =
simply 2-spheres, and the function
w (r, e , ¢) = w (r ) . Thus the slices w = constant are w of r is to be determined so that Rv = 1 67r Kp (r) ;
sinc e we are interested here in the region exterior to the ball, we shall not be concerned that p is not known explicitly as a function of r . For the entire V 3 sitting i n IR,4 , w e may again apply Gauss's equation ( 1 1 .79), where now the K 'S are the principal curvatures of V 3 C IR,4 . It is easy to compute the normal curvatures for this 3-dimensional analogue of a surface of revolution, and in Chapter 5 of [FrJ it is shown that exterior to the ball, w takes a parabolic form, yielding the Flamm paraboloid
w2
=
8m (r - 2m)
where
m
=
K
and
g rr
=
(
loa 47r r 2 p (r) dr
)
2m - '
1 - 7
a measure of the "total mass" of the ball of coordinate "radius" a. Thus V 3 carries spatial metric ( 1 - 2 m/r) l d r 2 + r 2 (de 2 + sin2 ed¢ 2 ) . In Problem 1 1 . 1 (2) i t was shown that U = ( 1 - 2m/r) I / 2 i s a solution to Laplace ' s equation in the spatial Schwarzschild metric, and, for large r, U is of the form U '"" is
the
-
322
RELATIVITY, TENSORS, AND CURVATURE
1 - m f r . Thus 1 - ( 1 - 2m/r) I / 2 i s a good candidate for the "correct" gravitati onal potential in the exterior region. As in Section 1 1 . I c, this suggests that -J=goo = 1 - U == ( 1 - 2m /r) I /2 and so goo = - ( 1 - 2m/ r ) . In [Fr] it is shown that this is in fact the solution demanded by the remaining Einstein equations. Thus in the external region We have the Schwarzschild solution
ds 2
=
-
(- )
2m ""7 d t 2
I
+
(1 - )
2m - 1 dr2 ""7
+
r2 (de 2
+
sin 2 ed¢2)
( 1 1 .86)
Problems
1 1 .5(1 ) Consider the classical electromagnetic field in
IR 3 , as in Section 3.5. Note th at
(�/\* 1;; E E VO/3 , and so differentiating with respect to ti me gives a j a t (0 M�;) == 20 /\ a* t;ja t. Li kewise, we may compute a j a t UJ.l /\ *�il ) . Show that for a fixe d compact region U of IR3 , we have =
•
1
,-
1
v
c �,
/\ * l" + (, h0 /\ *.()h =
- 41
1
·2 (0 l" /\ *. h l" /\ ' ( 1 1 .87) v J n av The integrand on the left-hand side is (8n)- 1 ( E2 + 82 ) 2: 0 and is the claSSical energy density of the field. Note that Iv (; 1 /\ Iv E J VO/3 Iv E p v voJ3 d 1 dt 8n
- -
-
C
/
=
•
<'
�
•
represents the rate at which the field does work on the charges in the cu rrent. Then (4n ) - 1 J� V (�1 /\ *�il 2 Ia u (4n ) - 1 E x B . dS is i nterpreted as the flux of energy th rough a u. Relativistically, energy is the same as mass. But the flux of mass through a surface is given classically by the su rface i ntegral of the momentum density. ( For example, in the case of a fluid with mass density p we have Ia u pv · dS = - dj d t Iv p vo/ 3 . ) Thus we may consider (4n ) - 1 E x B, the Poynting vector, to be the momentum density of the field. Equation ( 1 1 .87) is Poynting's theorem. 1 1 .5(2) I n the Riemannian case one puts 4> = ( goo ) 1 / 2 , but ( 1 1 .8 1 ) still holds. Show that =
(see ( 1 1 .39» . Since Vdet(g"fi ) dx1 /\ . . . /\ dx n- 1 is the "area" form d Sn-1 for Vn- 1 we may write for the fi rst variation of area
:!..- r This is the (n term?
1 1 .5(3) Prove ( 1 1 . 82) . 1 1 .5(4) Prove ( 1 1 .83) .
dt
) V(t)
d Sn - 1
=
_
- 1 ) -dimensional version
r 4> H d Sn-1 } V( t)
of (8.26) , but where is the boundary
CHAPTER
12
C u rvature and Topology : Synge 's Theorem
In Problem 8.3(7) it was shown that if M 2 is a closed surface in ]E. 3 then its curvature K and its "genus" g are related by 1 r _ KdS = 2 - 2g ( 1 2. 1 ) 2JT JM Thi s is the Gauss-Bonnet theorem. In particular, when M 2 is a (perhaps) distorted torus (i.e., a surface of genus 1 ), then (2JT ) - 1 iM KdS = O. Thus it is not possible to embed the torus in ]E.3 in such a way that its Gauss curvature is everywhere positive. This is not surprising; a few sketches of tori will "convince" one that there will always have to be saddle points somewhere. However, in Part Three, we shall see that ( 1 2. 1 ) is true even for an abstract Riemannian metric (without any question of an embedding in ]E. 3 ) . This i s an example of a global or topological result, relating the purely "infinitesimal" notion of curvature to the topological notion of the genus of the surface. In this brief chapter we will discuss a relation between curvature and the topological notion of simple connectivity, namely the theorem of J. L. Synge, one of the most beautiful results in global differential geometry of the twentieth century. In the process of proving Synge ' s theorem, we shall derive a formula, also due to Synge, for the second variation of arc length along a geodesic.
Figure 1 2.1
323
324
CURVATURE AND TOPOLO G Y : SYNGE'S THEOREM
I n the figure, w e have drawn a closed geodesic C, first on a surface with neg ative curvature and then on a positively curved sphere. If we consider only variati ons of C by smooth closed curves (where P = Q and the tangents match up at P ) , it is c lear from ( 1 004) that the first variation of arc length vanishes in both cases, the endpoint contributions cancel in the case of a closed geodesic. Still, we can shorten the equator C on the sphere by pushing it north ! We could say that in the "space Q S 2 of all smooth closed curves on S 2 ," the length functional L has first derivative 0 at the point representing the equator C but C does not yield a relative minimum for L . We s hall see, from Synge's formula, that in thi s case of positive curvature the second variation is negative for the variation pushing C north, explaining why this geodesic is unstable. (A slippery rubber band stretched along the equator would contract if disturbed slightly.) It seems evident that the equator in the negatively curved surface is stable, y ieldi n g an (absolute) minimum for L , and this will also follow from Synge's formula. 12. 1 . Synge's Formula for Second Variation
What does curvature have do with the stability of a geodesic?
12.1a. The Second Variation of Arc Length We first introduce a notation that will simplify the appearance of our calculations. Consider, as in Section 1 0. 1 b, the variation of arc length . We have the tangent vector fi eld T = ax/as and the variation field J = ax/aa, both defined along the 2-dimensional variational surface. We shall write with some misgivings
V'T J : =
V'J as
-
and
V'T V'JT : = aa
( 1 2.2)
even though T and J are defined only along the variational surface. We shall also write, for instance, V'T W rather than V' w las when W is a field defined along the variational surface. Thus Lemmas ( 1 0. 1 ) and ( 1 0.2) of Section 1 0. 1 then take the form
V'TJ = V'JT ( 1 2.3)
and
V'TV'JW - V' J V'T W = R (T, J)w We now return to our consideration o f arc length variation, started i n Section l O. l . We suppose now that the base curve Co , given b y a = 0 , i s a geodesic o f length L. Recall that the parameter s need be arc length only when a = O. We shall only be concerned with the case in which the fi rst variation vanishes, L' (0) = O. From ( 1 004) we see that this requires, in this case of a geodesic base curve, that
(J, T) Q = (J, T) p From the first equation of ( l OA) we have, in our new notation
L ' (a ) =
l L (T, T) - 1 /2 ( V'TJ, T)ds
SYNGE'S FORMULA FOR SECOND VARIATION
Then L" (a)
Since
II T 11 =
I
=
i \ - (T, T) -3j2 (V'TJ, T) 2
when a
L " (O)
=
Note th at
+ ( T, T) - 1 / 2 { (V'J V'TJ, T) 0, and since
=
+
325
(V'TJ , V'JT) } ]ds
V'TJ = V'JT
i L [- ( V'TJ, T) 2 + { (V'J V'TJ, T)
+
(V'TJ , V'TJ) } ]ds
where () is the angle between V'TJ and T. But this is simply the square of the area
II ( VTJ) /\ T 11 2 of the parallelogram spanned by these two vectors. Thus L " (O)
=
l L { (V'JV'TJ, T) + II (V'TJ) /\ T 1 1 2 } ds
( 1 2.4)
Look now at the first integrand
(V'J V'TJ, T) = (V'TV'JJ, T) + ( R(J, T)J, T) But
(V T V'JJ, T) = Cl / Cl s (V'JJ , T) - (V'JJ, V'TT) , and so L (V'TV'JJ, T)ds = (V'JJ, T)�
i
Equation ( 1 2.4) then becomes L " (O) =
( V'JJ, T) � +
i L { ( R (J, T)J, T) + II ( V'TJ) /\ T 11 2 }ds
The statement that the covariant Riemann tensor is skew in the first two i ndices translates
to the statement
( R (J, T)J, T) = - ( R (J, T)T, J) as one easily sees by expressing this in terms of components. Thus we finally have our principal formula, dating from the year 1 925 .
Synge's Formula (12.5): For a variation of a geodesic in which the first varia tion vanishes, (J, T) Q = (J, T) p, we have L { II ( V'TJ) /\ T 11 2 - ( R (J, T)T, J) }ds L" (O) = (V'JJ, T) � +
i
Note also that when the variation is orthogonal to the geodesic, that is, when (J, T) = 0 , then (V'TJ, T) = T (J , T) - (J, V'TT) = 0 , and I ( V'TJ) /\ T 11 2 becomes simply II VTJ 11 2 .
326
CURVATURE AND TOPOLOG Y : SYNGE'S THEOREM
Corollary (12.6) :
For a n othogonal variation of a geodesic w e have
Recall (from ( 1 1 .59)) that in a Riemannian manifold Mil , II A 11 2: 0 for all A and M has negative sectional curvature if ( R (J , T)T, J) is negative whenever T and J are linearly independent. Consider a geodesic C in such a space j oining distinct points p and Q . To see whether C locally minimizes arc length between P and Q we c onsider a variation J that vanishes at P and Q. Thus the endpoint contribution vanishes in Synge's formula. If J and T are not everywhere dependent along C the integral w il l be positive. If J = f ( s ) T along C , then the variation associated to J does not chan ge the curve C at all. From ( 1 2.5).
In a negatively curved Riemannian Mil , a nontrivial variation of a geodesic C joining distinct points P and Q yields LI/ (0) > 0 and so C is stable, that is, locally minimizes arc length.
Corollary (12.7):
In the case of a closed geodesic, J need not vanish at P = Q, but both T and J match up at P = Q , and so the first variation still vanishes. Furthermore, ('�7JJ, T) & = O. We conclude that LI/ (O) 2: 0, and = 0 only if J is a multiple of T along C ; this would simply move the geodesic into itself. Corollary (12.8) :
In a negatively curved Riemannian Mil , each closed geodesic
is stable. 12.1b. Jacobi Fields
We shall reconsider the case of distinct endpoints when the variation field J vanishes at the endpoints and is orthogonal to T. Then, as we have seen, ( V T J, T)
=
T (J, T) - (J, V T T) =
a (J, T) --
as
=
0
and so
Synge's formula then reads
( 12.9)
327
SYNGE'S FORMULA FOR SECOND VARIATION
V TJ) = T(J, V T J) - ( J, VT VTJ) and the first term integrates to 0 since (� But V TJ ' van i sh e s at the endpo ints. We then have J LI/ (O) =
-
foL { ( VT VTJ + R (J, T)T, J) } ds
( 1 2. 1 0)
to' va riations that vanish at the endpoints and are orthogonal to the geodesic.
Note th at if J is a Jacobi field, then LI/ (O) = O. Thus, from Problem 1 0. 1 (3),
rY
:tj,
th e geodesic C by a I -parameter family of geodesics passing the first and the second variations vanish !
if we through P and Q,
Thi s has the following consequence. (Our treatment will be very brief; for a more ] careful treatment see, e.g. , [Do, p. 423 .) = Xes) defined along a curve C , we can define at given any vector field X th First note
a
vari ation of C having variation vector given by X. There are many ways of forming s. For x (s , a) we may merely put x (s , a) = expx ( s ) aXes ) ; that is, x (s , a ) ari such v ation the geodesic starting a t x es) on C in the direction of X(s ) , and a t distance o n e t oin s p i th x es ) . from II aXes) II
Xes)
x(O)
Figure 1 2.2 Suppose that there is a nontrivial Jacobi field J along the geodesic C that vanishes at P and at :mme point p' between P and Q; we do not assume that J vanishes at Q . We call pI a conjugate point to P along the geodesic C .
;@ �- - - - - - - - -
p
p'
Figure 1 2.3
Q'
Q
328
CURVATURE AND TOPOLOG Y : SYNGE'S THEOREM
Using J w e may construct a variation o f the portion P p' o f C as before. Note that
different variations having the same variation vector J at ex = 0 will yield the same second variationformula ( 1 2 . 1 O) ! The varied curves Ca pass through p' and, by ( 1 2. 1 0),
have the same length, to second order, as the arc P p' of the base curve C . The varied curves Ca meet C = C (0) transversally i f ex is small enough; we se e this as follows. �e have already m� ntioned that a Jacobi field can e realized by varying C by geodeslcs Ca . If a geodeslc Ca were tangent to the geodeslc C at some point pi ' then Ca would coincide with C and so J == O. Thus Ca is transversal to C at P'. Let then P" be a point on Ca , and Q ' a point on C, that are so close that there is a unique minimal geodesic P" Q' j oining them. Then the geodesic arc P " Q' is strictly shorter than the broken arc P" p i Q ' . This says then that the curve of broken arcs p P" Q' Q is shorter than the original geodesic P pi Q' Q = p Q . The broken P P" Q' Q can then be smoothed off to yield a smooth curve that is again shorter than C. We h ave "shown" that
�
(12.11 ) : If a geodesic arc C contains a point pi conjugate to the be ginning point P in its interior. then C is not a minimizing geodesic; that is, C is not stable.
Theorem
Thus a geodesic cannot be minimizing after passing a point conjugate to the in itial point! In fact Marston Morse has shown the following (see [MD. Let us say that the point p i conjugate to P has (Morse) index A iff there are exactly A linearly independent Jacobi fields along C that vanish at both P and P'. (This makes sense since the Jacobi equation is linear in J.) Suppose that P{ , . . . , P; are exactly the conj ugate points to P that are between P and Q , and that P( has index A (i ) . We define the (Morse) index of C to be L i A U ) , the sum of the indices of all conjugate points pi interior to P Q . Then in a certain well-defined sense, there are essentially Li A(i) independent variations of
C that strictly decrease the arc length of C.
For example, consider on the n sphere the geodesic (great circle) C that starts at the north pole P , passes through a point Q on the equator, goes all the way around to P again, and continues on to the point Q .
p = p"
Q
Figure 1 2.4
329
CURVATURE AND SIMPLE CONNECTIVITY
point to P is the south pole p i ; the next and last is p I!
first conjugate earc length 2Jr ) . For the arc P p i there is an (n - 1 ) -dimensional family of great (�tc1es (parameterized by the equator S,, - I ) ; these yield an (n - I )-dimensional space � g at pi.' a.nd thus the index of the conj ugate point .p ' is � 1 . Jaco bi fi elds vanishi eld vanatlOns of the segment P p i P", and so p I! IS conjugate also e Yi cs s The geodesi 1 . Thus the Morse index of the geodesic P Q pi pI! Q is 2 (n - 1 ) ; p th dex n Th
��
��ere
=
P itself
-
wi
in
-
a re basica lly 2 n - 2 independent variations of C that decrease the length of C . Problem
1 2 .1 ( 1 ) Use ( 1 1 .82) and [ R(J , T)T]8
=
transformation
J
f-+
Tb R 8 bcd Jc Td to show that the Jacobi linear R(J , T)T
is self-adjoint.
12.2. Curvature and Simple Connectivity
How is positive curvature related to simple connectivity?
12.2a. Synge's Theorem (12.12) : Let M 2n be an even-dimensional, orientable manifold with positive sectional curvatures, K (X /\ Y) > O. Then any closed geodesic is unsta ble, that is, can be shortened by a variation.
Theorem
For example, the equatorial great circle on the round 2-sphere can be shortened by
pushing it north.
Let C , x = x e s ) , be a closed geodesic. We first claim that we can find a unit vector field J along C that is normal to C and parallel displaced along C. This is proved as follows. Since parallel translation around C will send the geodesic tangent T into itself, parallel translation around C will also take the (2n - I ) -dimensional plane of vectors normal to T into itself. Let T-L be the normal plane at x (0) . Parallel translation around C will give a map p : T-L -+ T-L . This map is linear since the differential equations of parallel translation are linear. We know that this map is an i sometry ; thus P is given by an orthogonal matrix, P T = P - I . P cannot reverse the orientation of T-L , for if it did, since T i s sent into itself, parallel translation would have reversed the orientation of the 2n -dimensional tangent space to M at x (O) , contradicting the assumption that M is orientable. Thus det P = + 1 . But the eigenvalues of P either are real or occur in complex conjugate pairs, and since there are 2 n - 1 of them, we conclude that there are an odd number of real eigenvalues. But each of these must be ± 1 , and yet det P = (the product of all the eigenvalues) = 1 . Thus PRO O F O F S Y N G E ' S T H E O R E M :
330
C U R V A T U R E A N D TOPOLOG Y : S Y N G E ' S T H E O R E M
there is a t least one eigenvalue A = + 1 . B ut this means that some normal vector J must be sent into itself under the parallel translation; J (s) is a normal parallel displaced vector along C ! We may then construct a variation of C by again considering the geode sic s tangent to the vectors J, that is,
x es, a)
: = expx(s) { a J (x (s )
}
By construction (3x /oa) (s , 0) = J (x (s) ; that is, this variation has J as its vari ation vector. Look at Synge's formula ( 1 2.9). The boundary term vanishes since we have a closed curve. Further, V T J = 0 since J is parallel displaced. Thus L " (O)
=
-
rL K (T
in
1\
J )ds
( 1 2. 1 3)
since T and J are orthonormal. We conclude that L" (0) < O . Since L ' (0) = 0 for the geodesic we conclude that such a variation would decrease the length of the curve for small a. 0 There are spaces with positive sectional curvatures. The usual paraboloid in lR3 has positive curvature, and any deformation of it, if sufficiently small, will also. Likewise for the unit sphere (which is compact). The unit sphere S" C lR/+ 1 has sectional curvatures all unity. To see this we use the Gauss equation 0 1 .60) applied to M = lR/+ 1 and V = S" . Since KM = 0 for M euclidean we have Kv (X, Y) = B (Y, Y) B (X, X) { B (X, y» ) 2 . For two orthogonal principal directions X = e I and Y = e 2 we would have KV (el ' e2 ) = K I K2 . But by symmetry, all principal curvatures for the round unit sphere must coincide, Ki = - 1 (using the outward-pointing normal) . Thus all sectional curvatures for the unit n -sphere are + 1 . For another example, consider the real projective n-space lRpll . This is the space resulting from the unit n -sphere when antipodal pairs are identified. Any tangent vector X to lRpn corresponds to a pair of tangent vectors, Y and - Y, to S" at antipodal points. These vectors have the same length, and thus there is no ambiguity in defining II X I I to be the length in the Riemannian S" of either of the tangent vectors ± Y "covering" X. This defines a Riemannian metric for lRP" . It should be clear that the 2: 1 projection (identification) map Jr : S" ---* lRP" is then a local isometry, and thus the Riemann tensors of the two spaces agree at corresponding points, if we use local coordinates in S" that result from pulling back local coordinates in lRP" (see Section 8 .Sb). Thus lRP" carries a Riemannian metric with sectional curvatures K = I again ! We have mentioned in Section 1 0. 2d that if a compact manifold i s not simply con nected, then among a free homotopy class of closed curves that cannot be shrunk to a point, there will be a shortest curve and it will be a closed geodesic. Thus we have Synge's theorem of 1 936.
compact, orientable, even-dimensional manifold with pos itive sectional curvatures is simply connected.
Corollary (12. 14) : A
C U R V A T U RE A N D
SIMPLE
331
CONNECTIVITY
12.2b. Orientability Revisited The example ]E. P " is especially interesting with regard to Synge's corollary because RP" is not simply connected! This can be seen as follows. An arc C' on S" going from the north to the south pole projects down to yield a closed curve C on ]E.P" since the north and south poles project to the same point (call it N) on ]E. p n . We claim that C cann ot be deformed to a point on ]E. p n . Let C be parameterized, x = x (t ) , with x(O) = x ( I ) = N. It should be clear that any deformation of C can be "covered" by a deform ation of C' on sn , using the identification. Under a deformation of C, the point N migh t move to another point Na , and then the covered curve C� would start at one of the two points on S" covering Na and end at its antipodal point Na . If C could be deformed to a point, then eventually we would have to cover thi s single point curve C 1 at NI by a whole arc on S" going from a point over NI to its antipode. This i s impossible since N I is covered only by two points on S" . 0 The fact that ]E. p 3 is not simply connected has the following application to mechanics ([A , p. 248]). -
Theorem (12.15) : A rigid body, free to rotate in ]E. 3 about a fixed point of the body and subject to any time independent potential field, has a periodic motion
for any sufficiently large total energy E.
C O M M E N T : A rigid body fixed at one point in ]R 3 has the rotation group
S O (3)
a s configuration space (see Section l . l d) . S O (3) is a n example o f a Lie group, and we shall study them in detail in this book, starting with Chapter 1 5 . One then looks at the phase space, i.e., the cotangent bundle to S 0 (3) . In Section 28 of Arnold's book it is shown how the classical moment of inertia tensor is used to relate (angular) velocities on the manifold S O (3) to (angular) momenta, and thence to define the kinetic energy on the phase space. Using the given potential field on SO (3) we can form the Jacobi metric of Theorem ( 1 0. 1 9). For sufficiently large total energy H = E, the Jacobi metric defines a Riemannian metric on all of SO (3) in which the geodesics represent the motions of the system. But S O (3) is topologically ]E.p3 (see 1 .2b, example vii). Since ]E.p3 is not simply connected, there exists a closed geodesic, and this corresponds to a periodic motion of the body. 0 Does the fact that ]E.p2/l is not simply connected contradict Synge's Corollary lRp 2n is compact, even-dimensional, and has positive sectional curvatures. Thus even dimensional projective spaces cannot be orientable ! This reaffirms the result of Problem 2.8( 1 ). Synge's method has another striking consequence for orientability. First note that if
I?
Mn is not orientable then there is some closed curve C that cannot be deformed to a point (in particular M is not simply connected!) and such that orientation is reversed on transporting an orientation around C. To see this, suppose that M is not orientable. Then it must be that transporting an orientation around some closed curve must lead to a reversal of orientation; otherwise it would be possible to transport an orientation
332
CURVATURE A N D TOPOLOGY : S Y N G E ' S THEOREM
uniquely from a given point t o every other point, implying that M was orientable . Let now orientation be reversed upon traversing a closed curve C. If we deform C slightly to a curve Cx , then, by continuity, orientation must be reversed also on traversi ng CO/ . Thus orientation would be reversed for every closed curve that is freely homotopic to C (see Section l O.2d). But if we could deform C to a point curve C I , where orientation cannot be reversed, we would have a contradiction. 0 Thus if Mil is not orientable, there is, from Section 1 0.2d, a closed geodesic C having the property that orientation is reversed upon traversing C and C is the shortest curve in its free homotopy class. In Problem 1 2. 1 ( 1 ) you are asked to prove the following: Corollary (12.16) : If M 21l+ 1
is a compact, odd-dimensional manifold with posi tive sectional curvatures, then M is orientable.
This shows that the odd-dimensional projective spaces are orientable.
Problem
1 2.2(1 ) Use Synge's method to prove Corollary ( 1 2 . 1 6) .
CHAPTER
13
Betti Numbers and De Rham 's Theorem When can we be certain that a closed form will be exact?
THE lack of simple connectivity is but one measure of topological complexity for a space. In this chapter we shall deal with others, the Betti numbers, and their relations with the potentials for closed exterior forms initiated in Chapter 5. This subject is a part of the discipline called algebraic topology. 13. 1 . Singular Chains and Their Boundaries
What does Stokes's theorem say for a Mobius band?
13.1a. Singular Chains
The standard (euclidean) p-simplex in lR,P is the convex set !1 p the p + 1 points
Po = (0, . . . , 0) , PI
=
C
lR, P generated by
( 1 , 0, . . . , 0) , . . . , Pp = (0, . . . , 0, 1 )
112
* Po
Po
It J
333
....____ . ....L... .; __ .. It J
Po
334
B E T T I N U M B E R S A N D DE R H A M ' S T H E O R E M
113
}-"""'-----::r-- 112
II I
Figure 1 3. 1 We shall write
A singular p-simplex in an n-manifold M n is a differentiable map
of a standard p-simplex into M.
Figure 1 3.2 Note that a singular simplex is a special case of a parameterized subset discus sed in Section 3 .4b. This is the natural object over which one integrates p-forms of M via the
335
SINGULAR CHAINS AND THEIR BOUNDARIES
p ull-b ack
We emphasize that we put no restriction on the rank of the map ap ; for example, the p which we shall also denote by aI" may be a single point of M. i mage of D. , at the kth face of D. p th e Not ) D. (kp_ 1
�
. - (Po, . . . , Pk , . . . , Pp) that is, the face opposite the vertex Pk. i s not a standard euclidean simplex, sitting as it does in � p instead of � p - I . We shall rather consider it as a singular simplex in � p . In order to
. _
do this we must exhibit a specific map
of D.p_ 1 into
�P, having the face as image. We do this in the following fashion. fk is p the unique affine map (i.e., a linear map followed by a translation of origin) of � - I into �p that sends Po --+ Po, . . . , Pk- I --+ Pk - I , Pk --+ Pk+I , . . . , Pp - I --+ Pp . il If a : D. p --+ Mil is a singular simplex of M and if ¢ : M --+ V r is a differentiable r map, then the composition ¢ 0 a : D. p --+ v defines a singular simplex of V . In particular a 0 fk : D. p _ 1 --+ M il defines a singular (p - I )-simplex of M, the kth face
singular p-simplex a . We define the boundary a D. p of the standard p-simplex, for p > 0, to be the formal sum of singular simplexes of the
= L( - I ) k D. (�_ 1
k
whereas for the O-simplex we put (Po, P2 ) + (Po, PI ) .
( 1 3. 1 )
a D. o = 0. For example, a (Po, PI , P2 )
=
(PI , P2 ) -
+
Figure 1 3.3
D. 2
=
(Po , PI , P2 ) is an ordered simplex; that is, it is ordered by the given ordering
of its vertices. From this ordering we may extract an orientation; the orientation of D. 2
336
BETTI NUMBERS AND DE RHAM'S THEOREM
is defined t o b e that o f the vectors e l = P I - Po and e 2 = P2 - Po · Likewise, each o f i ts faces is ordered by its vertices and has then an orientation. We think of the mi nus si gn in front of ( Po , P2 ) as effectively reversing the orientation of this simplex . Symbo lic ally,
a Figure 1 3.4
In this way the boundary of �2 corresponds to the boundary as defined in Sectio n 3 . 3 a, and, in fact, Stokes ' s theorem for a I -form a I in the plane says, for this � = � 2 ,
A similar result holds for � 3 . � 3 = ( Po , PI , P2 , P3 ) is an ordered simplex with orien tation given by the three vectors PI - Po , P2 - Po , and P3 - Po . As drawn, this is the right-hand orientation. a � 3 has among its terms the "roof" ( PI , P2 , P3 ) and it occurs with a coefficient + I . The orientation of the face + ( PI , P2 , P3 ) is determined by the two vectors P2 - PI , and P3 - PI , which is the same orientation as would be assigned in Section 3 . 3a. a fi p , as a formal sum of simplexes with coefficients ± I , is not itself a simplex. It is an example of a new type of object, an integer (p - I )-chain. For topological purposes it is necessary, and no more difficult, to allow much more general coefficients than merely ± I or integers. Let G be any abelian, that is, commutative, group. The main groups of interest to us are
G
= Z,
the group of integers
G G
= JR,
the additive group of real numbers
= Z2
=
Z/2Z,
the group of integers mod 2
The notation Z2 = Z/2Z means that in the group Z of integers we shall identify any two integers that differ by an even integer, that is, an element of the subgroup 2Z. Thus Z2 consists of merely two elements Z2 =
{O, I } where
o is the equivalence class of 0, ±2, ±4, . . . I is the equivalence class of ± 1 , ±3 , . . .
with addition defined by 0+0 = 0, 0+ f = f , I + I = O. This of course is inspired by the fact that even + even = even, even + odd = odd, and odd + odd = even. We usually write Z2 = {O, I } and omit the tildes-: Likewise, one can consider the group Zp = ZI p '/t ,
337
SINGULAR CHAINS AND THEIR BOUNDARIES
the group of integers modulo the integer p, where two integers are identified if their di fference is a multiple of p. This group has p elements, written 0 , 1 , . . . , p - 1 . We defi ne a (singular) p-chain on Mil , with coefficients in the abelian group G , to be a fin ite formal sum ( 1 3 .2) b.p -+ M, each with coefficient go E G. This formal defini tion means the following. A p-chain is a function c I' defined on all singular p si mpl exes, with values in the group G , having the property that its value is 0 E G for all but perhaps a finite number of simplexes. In ( 1 3 .2) we have exhibited explicitly all of the si mplexes for which cp is (possibly) nonzero and of singular simplexes
ap'
:
cl' (a/) = gs
We add two p-chains by simply adding the functions, that is,
(cl' + c� ) (ap )
:=
cp (al' ) +
c� (ap)
addition o n the right-hand side takes place in the group G . I n terms o f the formal sums we simply add them, where of course we may combine coefficients for any simplex that is common to both formal sums. Thus the collection of all singular p-chains of Mn with coefficients in G themselves form an abelian group, the (singular) p-chain group of M with coefficients in G, written Cp (M" ; G). A chain with integer coefficients will b e called simply a n integer chain. The standard simplex b. p may be considered an element of Cp (l�P ; Z) ; this p chain has the value 1 on b.1' and the value 0 on every other singular p-simplex. Then p a !:J. p = � k ( - I ) k !:J. (�;_ , is to be considered an element of C p_ ' (JR. ; Z) . A homomorphism of an abelian group G into an abelian group H is a map f : G -+ H that commutes with addition (i.e., f (g + g') = f (g) + f (g'» . On the left-hand side we are using addition in G ; on the right-hand side the addition is in H . For example, f : Z -+ JR. defined by f en) = n,Ji is a homomorphism. F : Z -+ Z2 , defined by F(n) = 0 if n is even and I if n is odd, describes a homomorphism. The reader should check that the only homomorphism of Z 2 into Z is the trivial homomorphism that sends the entire group into 0 E Z. Let F : Mn -+ v r • We have already seen that if a is a singular simplex of M then F o a is a singular simplex of V . We extend F to be a homomorphism F. : Ck ( M ; G) -+ Ck ( V ; G ) , the induced chain homomorphism, by putting The
F. (g ,
aI"
+ . ..
+ gr a/ )
:= g, (F 0
For a composition F : M il -+ v r and E
:
(E 0 F).
aI" ) + . . . + gr ( F 0 a/ )
vr -+ WI we have =
If a : !:J. p -+ M is a singular p-simplex, let its boundary
(p - I )-chain defined as follows. Recall that a b. I' is the integer (p
Lk ( - 1 ) k b. (�L , on !:J. p . We then define
aa
:=
a. (a b.) = L (_ l )k a. (b.(k) 1'- , ) k
( 1 3 .3 )
E. 0 F.
aa
be the integer - 1 ) -chain a !:J. I' = ( 1 3 .4)
338
BETTI NUMBERS AND DE RHAM'S THEOREM
Roughly speaking, the boundary of the image of t>.. i s the image of the bounda ry of Ll ! Finally, we define the boundary of any singular p-chain with coefficients in G by a
L gr a �
:=
L gr aa �
( 1 3. 5 )
By construction we then have th e boundary homomorphism a
:
Cp ( M ;
G)
�
Cp _ 1 ( M ;
G)
( 1 3.6)
v r and if c p = L: gr a� is a chain on M, then for the induced ch ai n F* c on V we have a ( F* c) = a L: g r F* a r = L: gr a ( F* a r ) = L: g r ( F 0 a r ) * (a t>.. ) == L: gr F* [a : (a t>.. ) ] = F* [L: gr a : (a t>.. ) ] = F* acp . Thus
If F
:
Mn
�
a
0
F*
=
F* 0 a
( 1 3.7)
(Again we may say that the boundary of an image is the image of the boundary.) We then have a commutative diagram F* Cp ( M ; G ) a t C p _ 1 ( M ; G)
�
�
C p ( V ; G) a t C p _ 1 ( V ; G)
F*
meaning that for each c E C p (M ; G) we have F* a c p = a F. (c p ) . Suppose we take the boundary of a boundary. For example, a a ( Po , PI , Pz) = a { ( p\ , Pz) - ( Po , Pz ) + ( Po , PI ) } = P2 - PI - ( Pz - Po ) + PI - Po = O. This crucial property of the boundary holds in general.
Theorem (13.8): aZ
=
a0a =0
Consider first a standard simplex t>.. p . From ( 1 3 . 1 )
PROOF:
a a t>.. p
=
'" L.,,(- l ) k a ( Po ,
=
L( - l l L ( - l ) j ( Po , . . . , Pj , . . . , Pk , . . . , PI' )
=
0
k
k
�
· · · , Pk , . . . , PI' )
j
(cancellation in pairs)
But then, for a singular simplex, a (a a ) a* a ( a t>.. )
= a* (O)
=
O.
D
=
a (a* ( a t>.. » , which, from ( 1 3 .7), is
13.1h. Some 2-Dimensional Examples cylinder Cyl is the familiar rectangular band with the two vertical edges brought together by bending and then sewn together. We wish to exhibit a speci fic i nteger 2 - chain
1. The
SINGULAR CHAINS AND THEIR BOUNDARIES
339
on Cyl. On the right we have the rectangular band and we have labeled six vertices. The Pz
Po
Figure 1 3.5 labels on the two vertical edges are the same, since the band is to be bent and the two edges are to be sewn, resulting in Cyl. On the band we have indicated six singular 2simplexes. We shall always write a singular simplex with vertices in increasing order. For example, ( Q " Q 3 , Q 4 ) is the singular simplex arising from the affine map of the plane into itself that assigns ( Po , P" P2) -+ ( Q " Q 3 , Q 4 ) . After the band is bent and sewn we shall then have a singular 2-simplex on Cyl that we shall again call ( Q " Q 3 , Q 4 ) . We have thus broken C y l u p into 2-simplexes, and w e have used enough simplexes so that any 1 - or 2-simplex is uniquely determined by its vertices. We wish to write down a 2-chain where each simplex carries the orientation indicated in the figure. Since we always write a simplex with increasing order to its vertices, we put C2
= ( Qo , Q " Q2 ) - ( Q o , Q " Q 3 )
+ ( Q " Q3 , Q 4 ) - ( Q3 , Q4, Q s )
+ ( Q2 , Q 4 , Q 5 ) + ( Q o , Q 2 , Q s )
Then
We write this as J C2 = B + C , where B = ( Q o , Q 3 ) + ( Q3 , Q s ) - ( Qo , Q s ) and C = ( Q2 , Q 4 ) - ( Q " Q 4 ) + ( Q " Q 2 ) . B and C are two copies of a circle, with opposite orientations; B is the bottom edge and C the top. Denote the seam ( Q o , Q2 ) by A , and omitting all other simplexes, we get the following symbolic figure.
c
B
or
340
BETTI N U M B E RS A N D DE R H A M ' S THEOREM
a{ B
}= Figure 1 3.6
C) C
B
C)
Note that in the lower figure the result is the same as would be obtained if we think of the cylinder as an oriented compact manifold with boundary, the boundary being then oriented as in Section 3. 3a. In the upper figure we have a rectangle with four sides. By denoting both vertical sides by the same curve A we are implying that these two sides are to be identified by identifying points at the same horizontal level . The bottom curve B and the top C, bearing different names, are not to be identified. As drawn, the bottom B, the top C, and the right-hand side A have the correct orientation as induced from the given orientation of the rectangle, but the left-hand A carries the opposite orientation. Symbolically, if we think of the 2-chain C2 as defining the oriented manifold Cyl, we see from the figure that a Cyl = B + A + C - A = B + C
2.
the same result as our calculation of a C2 given before with all of the simplexes. From the rectangular picture we see immediately that all of the "interior" I -simplexes, such as ( QJ , Q 4 ) , must cancel in pairs when computing a c2 . The Mobius band Mo. We can again consider a 2-chain C2
Figure 1 3.7 Note that the only difference is the right-hand edge, corresponding to the half twist given to this edge before sewing to the left hand edge; see Section 1 .2b (viii). This C 2 is the same as in the cylinder except that the last term is replaced by its negative - ( Qo, Q 2 , Q5 ) . We can compute a C 2 just as before, but let us rather use the symboliC rectangle with identifications.
341
SINGULAR CHAINS AND THEIR BOUNDARIES
c
A
Qo
A
B
Figure 1 3.8 The boundary of the oriented rectangle is now a Mo
=
B+A+C+A
=
B + C + 2A
This is surely an unexpected result! If we think of the Mobius band as an integer 2-
B
B
Figure 1 3.9 chain, as we did for the cylinder, then the "boundary," in the sense of algebraic topology, does not coincide with its "edge", that is, its boundary in the sense of "manifold with boundary." As a chain, one part of its boundary consists of the true edge, B + C , but note that although the point set B + C is topologically a single closed curve it changes its orientation halfway around. It is even more disturbing that the rest of the boundary consists of an arc A going from Q 2 to QQ, traversed twice, and located along the seam of the band, not its edge ! The reason for this strange behavior is the fact that the Mobius band is not orientable. It is true that we have oriented each simplex, j ust as we did for the cylinder, but for the cylinder the simplexes were oriented coherently, meaning that adj acent simplexes, having as they do the same orientation, induce opposite orientations on the I -simplex edge that is common to both. This is the reason that a C2 on the cylinder has no I -simplex in the interior; only the edge simplexes can appear in a C2 . On the Mobius band, however, the oriented simplexes ( QQ , Q I , Q2) and - ( QQ , Q 2 , Q s ) induce the same orientation to their common ( QQ , Q 2 ) = - A since these two 2-simplexes have opposite orientations ! This is a reflection of the fact that the Mobius band is not orientable. We shall discuss this a bit more in our next section.
342
BETTI NUMBERS AND DE RHAM'S THEOREM
We have defined the integral o f a 2-form over a compact oriented surface M 2 in Chapter 3 , but we mentioned that the integral i s classically defined by breaking up the manifold into pieces . This is what is accomplished by construction of the 2-chain C2 ! Let a ' be a I -form on the cylinder, oriented as in Example ( 1 ) . The integral of da over Cyl can be computed by writing Cyl as the 2-chain C2 . Applying Stokes 's theorem to each simplex will give
J"lcy,r
da ' =
r
lacy,
a'
=
r
lB+C
a'
=
r
lB
just as expected. However, for the Mobius band, written as
J"lMar
da ' =
r
la Ma
a'
=
r
lB+C+2A
a' =
r
lB
a' +
C2 ,
a' +
r
lc
r
lc
a'
a' + 2
r
lA
a'
This formula, although correct, is of no value. The integral down the seam i s not intrinsic since the position of the seam is a matter of choice. The edge integral is also of no value since we arbitrarily decide to change the direction of the path at some point. It should not surprise us that Stokes's theorem in this case is of no i ntrinsic value since the Mobius band is not orientable, and we have not defined the integral of a true 2-form over a nonorientable manifold in Chapter 3 . If, however, a ' were a pseudoform, then when computing the integral of d a ' over the Mobius C2 , Stokes's theorem. as mentioned in Section 3 .4d, would yield only an integral of a ' over the edge B + c . The fact that B and C carry different orientations is not harmful since the a that is integrated over B w i ll be the negative of the a that is integrated over C ; this is clear from the two simplexes ( Qo, Q " Q 2 ) and - ( Qo. Q 2 , Qs). 13.2. The Singular Homology Groups
What are "cycles" and "Betti numbers"?
13.2a. Coefficient Fields
In the last section we have defined the singular p-chain groups C p ( M il ; G) of M with coefficients in the abelian group G, and also the boundary homomorphism Given a map F : M il
--+
V r we have an induced homomorphism
and the boundary homomorphism a is "natural" with respect to such maps, meaning that
a 0 F*
=
F* 0
a
We also have a 2 = O. Notice the similarity with differentia/forms, as a takes the place of the exterior derivative d ! We will look at this similarity i n more detail later. Many readers are probably more at home with vector spaces and linear transforma tions than with groups and homomorphisms. It will be comforting to know then that in many cases the chain groups are vector spaces, and not just abelian groups.
THE SINGULAR HOMOLOGY GROUPS
343
An ab elian group G is a field, if, roughly speaking, G has not only an additive s tructure but an abelian multiplicative one also, with multiplicative identity element J , and th is multiplicative structure is such that each g =I- 0 in G has a multiplicative I I inverse g - such that gg- = 1 . We further demand that multiplication is distributive ect to addition. The most familiar example is the field 1ft of real numbers. The sp with re 2 integers do not form a field, even though there is a multiplication, since for example, 2 E 2 does not have an integer multiplicative inverse. On the other hand, 2 2 is a field . if we defin e multiplication by 0 . 0 = 0, 0 . I = 0 , and I I = I. In fact 2p is a field P is a prime number. In 25 , the multiplicative inverse of 3 is 2. ver whene When the coefficient group G is a field, G = K, the chain groups Cp (Mn ; K) b ecome vector spaces over this field upon defining, for each "scalar" r E K and chain
cp =
( L. g i a i ,, )
E
C,, (M" ;
K)
rc p
=
L (rgi )a ip
The vector space of p-chains is infinite-dimensional since no finite nontrivial linear combination of distinct singular simplexes is ever the trivial p-chain O. From ( 1 3.5) we see that when G = K is a field,
is a linear transformation. Finally, a notational simplification. When we are dealing with a specific space Mn and also a specific coefficient group G, we shall frequently omit M and G in the notation for the chain groups and other groups to be derived from them. We then write, for example, a : C" --+ C ,, - 1 .
13.2h. Finite Simplicial Complexes At this point we should mention that there is a related notion of simplicial complex with its associated simplicial (rather than singular) chains. We shall not give definitions, but rather consider the example of the Mobius band. We have indicated a "triangulation" of the band into six singular 2-simplexes in Example (2) of the last section. Each of these simplexes is a homeomorphic copy of the standard simplex, unlike the general singular simplex. Suppose now that instead of looking at all singular simplexes on Mo we only allow these six 2-simplexes and allow only I -simplexes that are edges of these 2-simplexes, and only the six O-simplexes (i.e., vertices) that are indicated. We insist that all chains must be combinations of only these simplexes; these form the "simplicial" chain groups C p ' Then Co (Mo; G) is a group with the six generators Qo , . . . , Q 5 ; C 1 has twelve generators ( Qo , Q d , ( Qo , Q 2 ) " ' " ( Q4 , Q s ) ; and C2 has the six given triangles as generators. If we have a field K for coefficients, then these chain groups become vector spaces of dimension 6, 1 2, and 6, respectively, and the simplexes indicated become basis elements. In terms of these bases we may construct the matrix for the boundary linear transformations a : C p --+ Cp _ l . For example a ( Qo , Q I ) = Q I - Q o tells us that the 6 by 1 2 matrix for a : C I ( Mo; 1ft) --+ Co (Mo; lR) has first column ( - I , 1 , 0, 0, 0, ol . The simplicial chain groups are of course much
344
BETTI NUMBERS AND DE RHAM'S THEOREM
smaller than the singular ones, but i n a sense to b e described later, they already C on tain the essentials, as far as "homology" is concerned, in the case of compact manifol ds .
13.2e. Cycles, Boundaries, Homology, and Betti Numbers Return to the general case of singular chains with a coefficient group G . We are go ing to make a number of definitions that might seem abstract. In Section 13.3 we shall
consider many examples. We define a (singular) p-cycle to be a p-chain zp whose boundary is O. The collection
of all p-cycles,
Zp (M ; G) : = {z p =
E
C p l azp = O}
ker a : Cp
-+
( 1 3. 9)
Cp_ 1
that is, the kernel a - I (0) of the homomorphism a , is a subgroup of the chain group Cp (called naturally the p-eycle group). When G = K is a field, ZI' is a vector subspace of C 1" the kernel or null spa ee of a , and in the case of a finite simplicial complex this nullspace can be computed using Gauss elimination and linear algebra. We define a p-boundary f3p to be a p-chain that is the boundary of some (p + 1 ) chain. The collection of all such chains
Bp (M; G)
: =
(f3p
= 1m
E
Cp l f3,)
a : Cp+ 1
= -+
for some Cp + 1 E Cp+d
aCp+ l ,
( 1 3. 10)
Cp
the image or range of a , is a subgroup (the p-boundary group) of C p ' Furthermore, af3 = a ac = 0 shows us that Bp C Z" is a subgroup of the cycle group. Consider a real p-chain c p on M" , that is, an element of C p C M ; JR) . Then cp = 2:: bi a/) , where bi are real numbers. If is a p-form on M, it is natural to define
aP
( 1 3. 1 1)
Then
( 1 3. 1 2) We shall mainly be concerned with integrating closed forms, daP = 0, over p-cycles zp. Then if zp and z'p differ by a boundary, z - z' = aCp + l , we have
l a P - 1 aP = 1 a P Jaer aP i da z
Z'
.:;: - z '
=
=
c
" =
0
( 1 3. 1 3)
Thus, as far as closed forms go, boundaries contribute nothing to integrals. When integrating closed/orms, we may identify two cycles if they differ by a boundary. This identification turns out to be important also for cycles with general coefficients, not just real ones. We proceed as follows. If G is an abelian group and H is a subgroup, let us say that two elements g and g' of G are equivalent if they differ by some element of H ,
g ' '" g iff g ' - g
=
h
E
H
THE SINGULAR HOMOLOGY GROUPS
345
Someti mes we will say g ' = g mod H . The set of equivalence classes is denoted by / H, and read G mod H. If g E G we denote the equivalence class of g in G / H by G [ g ] or som etimes g + H. Such an equivalence class is called a coset. Any equivalence cl ass [ ] E G / H is the equivalence class of some g E G , [ l = [g l ; this g is called a representative of the class but of course [g 1 = [g + h 1 for all h E H . Two equivalence cl asses can be added by simply putting [g + gil : = [g 1 + [g il . In this way we make G / H itself into an abelian group, called the quotient group. This is exactly the procedure we foll owed when constructing the group Z 2 = Z/2Z of integers mod 2 . We always have a map :rr : G -+ G / H that assigns to each g its equivalence class [g ] == g + H. :rr is, by construction, a homomorphism. When G is a vector space E, and H is a subspace F , then E / F is again a vector space. If E is an inner product space, then E / F can be identified with the orthogonal complement F 1. of F and :rr can be identified with the orthogonal projection into the F
E
v
�
Fl.
, 0'
---.:... , __......1. ..; ' _____ ElF 7l" V =
[v]
Figure 1 3. 1 0
subspace F 1. . If E does not carry a specific inner product, then there is no natural way to identify E / F with a subspace of E ; any subspace of E that is transverse to F can serve as a model, but E / F is clearly more basic than these non unique subspaces . Return now to our singular cycles. We say that two cycles z p and z;, i n Zp (M ; G) are equivalent or homologous if they differ by a boundary, that is, an element of the subgroup B p (M ; G) of Z p (M; G). In the case of the cycles Z p and the subgroup B p , the quotient group is called the pth homology group, written HI' (M; G)
H1" (M · G) :=
Z I' (M ·, G) Bp (M; G)
( 1 3 . 1 4)
When G = K is a field, Z p ' Bp ' and HI' become vector spaces. We have seen that Z and B are infinite-dimensional, but in many cases HI' is finite-dimensional ! It can be shown, for example, that this is the case if M" is a compact manifold. Before discussing this, we mention a purely algebraic fact that will be very useful.
Theorem (13.15): If
346
B E T T I N U M B E R S A N D DE R H A M ' S T H E O R E M
homomorphism of the quotient groups
G,
-
H,
�
G2
-
H2
The composition of the homomorphisms
n
We then have the following topological situation. If Mn is a compact manifold, there is a triangulation of M by a finite number of n -simplexes each of which is diffeomor phic to the standard n-simplex. This means that M is a union of such n -simplexes and any pair of such simplexes either are disj oint or meet in a common r-subsimplex (vertex, edge, . . . ) of each. (We exhibited a triangulation explicitly for the Mobius band in Section 1 3 . 1 b). These simplexes can be used to form a finite simplicial complex, for any coeffi cient group G , just as we did for the Mobius band. Since C1" Z p , and R p are then finitely generated groups, so is HI' : = Z p l R p . Now any simplicial cycle can be considered a singular cycle (i.e., we have a homomorphism from Z I' to Z 1' ) and this homomorphism sends R p to B p . Thus we have an induced homomorphism of the simplicial homology class HI' to the singular homology class Hp . It is then a nontrivial fact that for compact manifolds HI' = HI' ; that is, the pl h singular homology group is isomorphic to the p llz simplicial homology group ! (A homomorphism is an isomorphism if it is 1 : 1 and onto.) In particular the singular homology groups are also finitely generated (even though the singular cycles clearly aren' t) and if G is a field K , HI' i s finite-dimensional. When G is the field of real numbers, G = JR , the dimension of the vector space HI' is called the plh Betti number, written bp = b p (M)
( 1 3 . 16) b p ( M ) : = dim Hp ( M ; JR ) In words, b p is the maximal number of p-cycles on M, no real linear combination of which is ever a boundary (except for the trivial combination with all coefficients 0). Let F : Mn � V' be a map. Since, from ( 1 3.7), F* commutes with the boundary a, we know that F* takes cycles into cycles and boundaries into boundaries. Thus F* sends homology classes into homology classes, and we have an induced homomorphism
( 1 3. 17 ) Finally, we can see the importance of the homology groups. Suppose that F : Mn � V" is a homeomorphism, then we have not only ( 1 3 . 1 7) but the homomorphism F -; I : Hp ( V ; G ) � Hp ( M , G ) induced by the inverse map, and it is easily seen that these two homomorphisms are inverses. Thus F* is an isomorphism; homeomorphic manifolds have isomorphic homology groups. We say that the homology groups are topological invariants. Thus if we have two manifolds M and V , and if any of their homology groups differ, for some coefficients G , then these spaces cannot be homeomorphic ! Unfortunately, the converse is not true in general; that is, nonhomeomorphic manifolds can have the same homology groups.
HOMOLOGY GROUPS OF FAMILIAR MANIFOLDS
347
1 3.3. Homology Groups of Familiar Manifolds
Is projective 3-space diffeomorphic to the 3-torus?
AnY point
13.3a. Some Computational Tools
p E Mn can be considered as a O-chain. By the definition of the boundary = 0, and each point is a O-cycle. or a p operat A smooth map C : [0, 1 ] -+ M is a curve in M; the image is compact since the e of a compact space (e.g., the unit interval) under a continuous map is again mag i compact (see Section I .2a) . C of course can be considered a singular I -simplex, and we have a C = C ( 1 ) - C (0) . If g E G, the coefficient group, then a (g C) is the O-chain gC ( ! ) - g C (O) . Suppose that C : [a , b] -+ M is a piecewise smooth curve. We may then break up the [a , b] into subintervals on each of which the map is smooth. By reparameteriz erval int curve on each subinterval, we may consider the mappings of the subintervals as ing the defining singular simplexes. We may then associate (nonuniquely) to our original curve a singular I -chain, associating the coefficient + I to each of the I -simplexes. The bound ary of this chain is clearly C (b) - C (a ) , the intermediate vertices cancelling in pairs.
Figure 1 3. 1 1 A manifold M" is said to be (path-)connected if any two points p and q can be joined by a piecewise smooth curve C : [0, 1 ] -+ M ; thus C (O) = p and C ( 1 ) = q . This curve then generates a I -chain, as in Figure 1 3 . 1 1 . But then a c = q - p. Likewise a (gC) = gq - gp , where gC is the I -chain that associates g E G to each of the I -simplexes. This shows that any two O-simplexes with the same coefficient, in a connected manifold, are homologous. Also, since a I -chain is merely a combination C = L g i Ci , ac = L { g i qi - gi Pi } , we see that no multiple gp of a single point is a boundary, if g #- 0. Thus any particular point p of a connected space defines a O-cycle that is not a boundary, and any O-chain is homologous to a multiple gp of p. We then have Ha (M " ;
G)
=
Gp for M connected
( 1 3 . 1 8)
meaning that this group is the set { gp i g E G } . For example, Ha (M" ; Z) is the set {a, ±p , ±2p , ±3p, . . . } and Ha (M" ; JR ) is the I -dimensional vector space consisting of all real multiples of the "vector" p. This vector space is isomorphic to the vector space JR, and we usually write Ha (M" ; JR ) = R In particular, a connected space has
348
BETTI NUMBERS AND DE RHAM'S THEOREM
= 1 . I f M is not connected, but consists of k connected piec es , then Ho (M" ; JR. ) = JR.P I + JR.P2 + . . . + JR.Pb where Pi is a point in the i th piece. In this case bo (M) = k. (We should mention that in topology there is the notion o f a connected space; M is connected if it cannot be written as the union of a pair of disjoint open sets. This is a weaker notion than pathwise connected, but for manifolds the two definitions agree.) Next, consider a p-dimensional compact oriented manifold V P without boundary. By triangulating V P one can show that V P always defines an integer p-cycle, which we shall denote by [ Vp ] ' For example, consider the 2-torus T 2 .
ot h Betti number bo
Qo
r------,
A
B
Qo
Qo
B
A
L..-______--------'
Qo
Figure 1 3. 1 2 If we associate the integer + I to each o f the eighteen indicated oriented 2-simplexes, we get a chain [ T 2 ] , for example, [ T 2 ] ( Q s , Q7 , Q s ) = - I .The boundary of this chain is clearly 0 a [T 2 ] =
A
+
B
-
A
-
B
=
0
and this same procedure will work for any compact orientable manifold. On the other hand, consider a nonorientable closed manifold, the Klein bottle K 2 . Thi s surface cannot be embedded in JR. 3 but we can exhibit an immersion with self intersections. This is the surface obtained from a cylinder when the two boundary edges are sewn together after one of the edges is pushed through the cylinder. Abstractly, in
Figure 1 3. 1 3
HOMOLOGY GROUPS OF FAMILIAR MANIFOLDS
349
termS of a rectangle with identifications, we have the following diagram; note especially the directions of the arrows on the circle B . _-------, Q o A Qo
B
B
Qo A 1-_________-----..... Qo
Figure 1 3. 1 4 Orient each o f the triangles as indicated and give to each oriented triangle the coef ficient + 1 , yielding a singular 2-chain [ K 2] . Now, however, we have a [ K2]
=
A+B
-
A
+ B = 2B i= 0
[K2] is not a cycle, even though the manifold has no boundary, that is, edge. This is a reflection of the fact that we have not been able to orient the triangles coherently; the Klein bottle is not orientable. Note another surprising fact; the I -cycle B = ( Q o , Q 2 ) + ( Q 2 , Q 3 ) - ( Qo , Q 3 ) is not a boundary (using Z coefficients) but 2 B is, 2 B = a [ K 2 ] ! Note also that if we had used real coefficients then B itself would be a boundary since then B = a ( l /2) [ K 2] , where this latter chain assigns coefficient 1 12 to each oriented 2-simplex. Furthermore, if we had used Z2 coefficients, then [ K 2 ] would be a cycle, since 2B = 0 mod 2. All these facts give some indication of the role played by the coefficient group G . The following theorem in algebraic topology, reflecting the preceding considerations, can be proved.
Theorem (13.19) : Every closed oriented submanifold VI' C Mil defines a p cycle g[ VI'] in HI' (Mil ; G ) by associating the same coefficient g to each oriented p-triangle in a suitable triangulation of V 1' . Th us a p-cycle is a generalization of the notion of a closed oriented submanifold. Rene Thom has proved a deep converse to ( 1 3 . 1 9) in the case of real coefficients.
Thorn's Theorem (13.20): Every real p-cycle in M il is homologous to a finite formal sum I: ri v/ of closed oriented submanifolds with real coefficients.
350
B E T T I N U M B E R S A N D DE R H A M ' S T H E O R E M
Thus, when looking for real cycles, w e need only look at submanifolds. Our next computational tool is concerned with deformations. In Section l O . 2d we discussed deforming closed curves in a manifold. In a similar fashion we can deform submanifolds and more generally p-chains. We shall not go into any details, but merely mention the
Deformation Theorem (13.21): If a cycle z p is deformed into a cycle z � . then z Ip IS• homo logous to z p , Z I p '" z p .
Figure 1 3. 1 5 This follows from the fact that in the process of deformi ng z p into z � one sweeps out a "deformation chain" c p + J such that ac p + ! = z ;, - z p . Our final tool is the following. For a closed n-manifold Mil , we know from Section 1 3 .2c that the singular homology groups are isomorphic to the simplicial ones. But in the simplicial complex for Mil there are no simplexes of dimension greater than n . Thus,
Hp (M" ; G)
=
0
for p
>
n
( 1 3 .22)
13.3b. Familiar Examples
1 . S" , the n -sph e re , n > 0. HO (S" ; G) = G since S" is connected for n > O. Since sn is a 2-sided hypersurface of lR" + ! it is orientable, and since it is closed we have Hn (s n ; G) = G . If z p is a p-cycle, ° < p < n, it is homologous to a simplicial cycle in some triangulation of S" . (The usual triangulation of the sphere results from inscribing an (n + I ) -dimensional tetrahedron and projecting the faces outward from the origin until they meet the sphere . ) In any case, we may then consider a z p that does not meet some point q E S" . We may then deform z p by pushing all of sn - q to the antipode of q , a single point. Z p is then homologous to a p-cycle supported on the simplicial complex consisting of one point. But a point has nontrivial homology only in dimension O. Thus z p '" ° and Ho (S " ;
Hp (sn ;
G)
=
G
G)
=
0,
The nonvanishing Betti numbers are bo 2. T 2 , the 2-torus. Ho = H2 = G .
=
=
H,, (S " ; G ) for p :f.: 0, n
1 = bll •
( 1 3 .23 )
351
HOMOLOGY GROUPS OF FAMILIAR MANIFOLDS
_-------...,
Qo
A
Qo
Qo
---,l Q , B
B
Qo
1------__-----
A
..... Q o
Qo
Figure 1 3. 1 6 Orient each 2-simplex a s indicated, as w e did i n Section 1 3 .3a. o [ T 2 ] = A + B A - B = 0, confirming that we have an orientable closed surface. Any I -cycle can be pushed out to the edge. It is clear that if we have a simplicial I -cycle on the edge that has coefficient g on, say, the simplex ( Q I , Q 4 ) , then this cycle will also have to have coefficient g on ( Qo, Q I ) and -g on ( Qo , Q 4 ) , since otherwise it would have a boundary. Thus a I -cycle on the edge will have the coefficient g on the entire I -cycle ' A. Likewise it will have a coefficient g on the entire I -cycle B . It seems evident from the picture, and can indeed be shown, that no nontrivial combination of A and B can bound. (For example, in Figure 1 3 . 1 6 we may introduce the angular coordinate e going around in the A direction. Then JA "de" #- 0 shows that A does not bound as a real i-cycle.) We conclude that
Ha (T 2 ; G ) = G = H2 ( T 2 ; G) HI (T 2 ; G ) = G A + G B
In particular,
HI (T 2 ; JR. ) = JR.A
+
JR.B i s 2-dimensional, bo
Qo
Figure 1 3. 1 7
( 1 3 . 24 ) = b2
= 1, bl = 2.
352
B E T T I N U M B E R S A N D DE R H A M ' S T H E O R E M
I n the figure w e have indicated the basic I -cycles A and B . The cycle B ' is homol ogous to B since B - B' is the boundary of the cylindrical band between them. The cycle C is homologous to 0 since it is the boundary of the small disc. 3. K 2 , the Klein bottle. Look at integer coefficients.
A
Qo
B
B
A
Figure 1 3. 1 8
= A + B - A + B = 2 B #- 0, the Klein bottle is a closed manifold but is not orientable. Again any I -cycle can be pushed out to the edge, Z I r A + s B , r and s integers. Neither A nor B bound, but we do have the relation 2B O. A satisfies no nontrivial relation. Thus A generates a group ZA and B generates a group with the relation 2B = 0; this is the group Z2 . Hence
Ho
=
Z but H2 = 0 since a [K 2 ]
�
�
Ho( K 2 ; Z)
=
Z, H2 (K 2 ; Z)
H I (K 2 ; Z)
=
ZA
+
=
0
( 1 3 .25)
Z2 B
If we used lR coefficients we would get ( 1 3.26)
since now B = a ( l /2) [K 2 ] bounds. Thus bo = 1 , b l = 1 , and b2 = O. 4. lR P 2 , the real projective plane. The model is the 2-disc with antipodal identifica tions on the boundary circle. The upper and lower semicircles are two copies of the same
HOMOLOGY GROUPS OF FAMILIAR MANIFOLDS
353
Qo
Figure 1 3. 1 9 closed curve A . One should triangulate ffi. P 2 but we shall not bother to indicate the trian gles. Orient all triangles as indicated. Clearly HO (ffi. p 2 ; .'2:) = .'2:. Since 3 [ffi. p 2 ] = 2 A , we see that the real projective plane is not orientable and H2 (ffi. p 2 ; .'2:) = O. A is a I -cycle and 2A O. �
HO (ffi. p 2 ; .'2:) = .'2: .
( 1 3 .27)
H I (ffi. p 2 ; .'2:) = .'2:2 A
With real coefficients ( 1 3 .28)
and bo = I , b l = 0, and b 2 = O. ffi.p2 has the same Betti numbers as a point! 5. ffi.p 3 , real projective 3-space. The model is the solid ball with antipodal indenti fications on the boundary 2-sphere. Note that this makes the boundary 2-sphere into a projective plane !
..
A
... . . . .
.......
.
.. . . . .... . . . . ...... ..... . . .
Qo
Qo
Figure 1 3.20
354
B ETTI NUMBERS AND DE RHAM'S THEOREM
Orient the solid ball using the right-hand rule. The upper and lower hemisphe re s B2 are two copies of the same projective plane lR P 2 . Orient the identified hemispheres B 2 as indicated. Note that the orientation of the ball (together with the outward normal) induces the given orientation in the upper hemisphere but the opposite in the low er. Orient the equator A as indicated. Ho (lRP3 ; lR) = R A is a I -cycle, but a B2 = 2 A and so H I (lRp 3 ; lR) = O. B 2 is not a cycle since a B 2 = 2 A #- 0, and so H2 (lRP\ lR) = O. a [lR p 3 ] = B 2 - B 2 = O. Hence lR p 3 is orientable (see Corollary ( 1 2. 1 4)) and H3 (lR p3 ; lR) = R
HO (lRp 3 ; lR)
lR = H3 (lRp 3 ; lR)
=
( 1 3.29 )
all others are 0
lR p 3 has the same Betti numbers as
S 3 ! See Problem 1 3 .3 ( 1 ) at this time.
6 . T 3 , the 3-torus. The model is the solid cube with opposite faces identified.
A
�
A
B
C
B
~ 0
B
B
A
A
Figure 1 3.21 Note that the front, right side, and top faces (which are the same as the back, left side, and bottom faces) p 2 , S 2 , and T 2 become 2-toruses after the identification. Orient the cube by the right-hand rule. This induces the given orientation as indicated for the drawn faces but the opposite for their unlabeled copies. Orient the three edges A , B, and C as indicated. A , B, and C are I -cycles. p 2 , S 2 , and T 2 have 0 boundaries just as in the case of the 2-torus. a [T 3 ] = p 2 + S 2 + T 2 - p 2 - S 2 - T 2 = 0 and so T3 is orientable. We have
( 1 3 .24')
Using real coefficients we would get bo
=
1 , bl
=
3
=
b2 , b 3
=
1.
DE RHAM'S THEOREM
355
Problems
p3
1 3 . 3( 1 ) Compute the homology g roups of lR. with Z coefficients. 1 3. 3(2) A certain closed s u rface M 2 has as model an octagon with the indicated iden tifications on the boundary. Note carefully the di rections of the arrows.
c Figure 1 3.22 Write down Hi ( M 2 ; su rface orientable?
G)
for
G
= lR.
and
G
=
Z. What are the Betti numbers? Is the
13.4. De Rham '8 Theorem
When is a closed form exact?
13.4a. The Statement of De Rham's Theorem In this section we shal l only be concerned with homology with real coefficients lR for a manifold Mn . The singular chains C ' cycles Z p ' and homology groups Hp then form P real vector spaces. We also have the real vector spaces of exterior differential forms on Mn .
A P : = all (smooth) p-forms on M FP : = the subspace of all closed p-forms EP := the subspace of all exact p-forms
We have the linear transformation a : C P � C p- l , with kernel Z p and image Bp- l yielding Hp = Z p / Bp . We also have the linear transformation d : A P � A P + l with kernel FP and image EP+ l C FP+ l , from which we may form the quotient �R P : =
FP
-
Ep
= (closed p-forms)/(exact p-forms)
( 1 3 .30)
356
B E T T I N U M B E R S A N D DE R H A M ' S T H E O R E M
the d e Rham vector space. �J� p is thus the collection o f equivalence classes o f clo sed p_ forms; two closed p-forms are identified iff they differ by an exact p-form. De Rh am ' s theorem ( 1 93 1 ) relates these two quotient spaces as follows. Integration allows us to associate to each p-form f3l' on M a linear function al I f3P on the chains C p by If3l' (c) = Ie { F . We shall, however, only be interested in this lin ear functional when f3 is closed, df3 P = 0, and when the chain c = Z is a cycle, d Z ::::: O. We thus think of integration as giving a linear transformation from the vector sp ace of closed forms F I' to the dual space Z; of the vector space of cycles
I . FP •
-+
by
(I f3P) (z)
:=
Z*P ( 1 3 .3 1 )
1 f3 P
Note that Iz+a e f3P = Iz f3P , since f3 is closed. Thus I f3P can be considered as a linear functional on the equivalence class of Z mod the vector subspace B p . Thus ( 1 3.3 1 ) really gives a linear functional o n HI'
I : Ff'
-+
H;
Furthermore, the linear functional I f3 P is the same linear functional as I (f3P + daP- 1 ), since the integral of an exact form over a cycle vanishes. In other words, ( 1 3 .3 1 ) really defines a linear transformation from FP / EP to H;, that is, from the de Rh am vector space to the dual space of HI" This latter dual space is commonly called the ih cohomology vector space, written H I' ( 1 3.32)
Thus ( 1 3 .33)
In words, given a de Rham class b E HiP , we may pick as representative a closed fonn f3 p . Given a homology class J' E HI" we may pick as representative a p-cycle z p . Then I (b) (J') : = I: f3 1' , and this answer is independent o f the choices made. Poincare conjectured, and in 1 93 1 de Rham proved
de Rham's Theorem (13.34): I : �iU' -+ H p (M; �) is an isomorphism. First, I is onto; this means that any linearfunctional on homology classes is of the form I f3 P for some closed p-form f3. In particular, if HI' is finite-dimensional, as it is when Mil is compact, and if Z p( I ) , . . . , Z p(b)
b = the pth Betti number
is a p-cycle basis of HI" and iflr l , . . . , lrb are arbitrary real numbers, then the re is a closed form f3 P such that
1. z. p (1
)
f3 P
= lr;
,
i=
I,
.
.., b
( 1 3 .35)
357
DE RHAM'S THEOREM
Secon d, I is (JP is exact,
I: I ;
this means that if I (fJl' ) (z,, )
=
r f3" = ° for all cycles z,,' then
jo r someform a ,, - I . The
number :rri in ( 1 3 .35) is called the period of the form f3 on the cycle z ,,(i) . Thus
a closed plorm is exact iff all of its periods 011 p-cycles vanish. A finite-dimensional
vector space has the same dimension as its dual space. Thus
Corollary (13.36): If Mn is compact, then dim �J1P = b", the pth Betti l1umba Thus bp is also the maximal number of closed p-forms on MI1 , no linear combi nation of which is exact. The proof of de
Rham's theorem is too long and difficult to be given here. Instead, we shall illustrate it with two examples. For a proof, see for example, [Wa] .
13.4h. Two Examples 1.
T 2 , the 2-torus. T2 is the rectangle with identifications on the boundary.
A 21l' 1-----.,
B
B
-------..1...--
o
A
21l'
e
Figure 1 3.23
�lO consists of closed O-forms, that is, constant functions, with basis f = ffi l
1.
consists of closed i -forms. Certainly de and d¢ are closed I -forms and these are not really exact since e and ¢ are not globally defined functions, being multiple valued. Since H I (T2; �) = �A + �B, A and B give a basis for the I -dimensional real homology. But then fA de 12:rr = 1 , fn de 12:rr = 0, fA d¢l2:rr = 0, and 18 d¢ 12:rr = i , show that de 12:rr and d¢ 12:rr form the basis in �J( I = H I = H * that
is dual to the basis A , B !
I
358
BETTI NUMBERS AND DE RHAM'S THEOREM
�R 2 consists of closed 2-forms, but of course all 2-forms o n T2 are closed. d e 1\ d¢ is closed and has period J J� ] de 1\ d
2. The suiface of genus 2.
F i g u re 1 3.24
�JLO has generator the constant function f = l . ,R2 has generator any 2-form on M 2 whose integral over [M 2 ] is different from 0, for example, the area 2-form in any
Riemannian metric . We need then only consider Ul 1 . This surface can be considered as an octagon with identifications on the edges. This can be seen as follows.
F i g u re 1 3.25
In the first step we merely narrow the neck. In the second step we cut the surface i n two along the neck; the result is a left and a right torus, each with a disc removed, the disc in each case having the original neck circle E as the edge. Of course these two curves must be identified. We now represent each punctured torus as a rectangle with identifications and with a disc removed. All vertices are the same Qo.
359
DE RHAM'S THEOREM
A
D
c E
c
B E A
D
Figure 1 3.26 We now open up the punctured rectangles
Figure 1 3.27 where again all vertices are the same Q o . Finally we may sew the two together along the seam E, which now disappears
Figure 1 3.28 leaving the desired octagon with sides identified in pairs. (Note that this is not quite the surface that appeared in Problem 1 3 .3(2) because of the identifications on the sides B . ) From this diagram the first homology i s clearly H I ( M2 ;
JR. ) = JR.A + JR. B + JR.C + JR.D, hi
=
4
360
BETTI NUMBERS A N D D E RHA M ' S THEOREM
We now wish to exhibit the dual basis in �,� I . Suppose, for instance, w e wi sh to construct a closed I -form whose period on A is I and whose other periods van i sh . Take a thin band on M 2 stretching from the interval pq on A to the same points on the identified other copy of A .
q
A
Qo / = 0 here
p
q
-1---- / = I here
q
Qo
Figure 1 3.29 Define a "function" f on M 2 as follows. Let f = 0 to the "left" of the band, let f = I to the "right" of the band, and let f rise smoothly in the band to interpolate. This is not really a function on M 2 since, for example, the side B is in both the left and the right regions. It does define a multiple-valued function ; we could have f starting with the value 0 to the left, and f increases by I every time one crosses the band from left to right. Although f is multiple-valued, its differential df is a well-defined I -form on all of M 2 ! By construction we have
1 df = I ,
and
f
} B or C or D
df
=
0
We have then exhibited the dual I -form to the class A . Using other bands we can construct the remaining dual basis forms for �,� I . Problem
2 � T2 of a sphere i nto a torus has degree O. : 8 "d¢" on T2 and show pull-bacK is exact.
1 3.4(1 ) (i) Show that every map F H i nt: Use "de"
/\
(ii) Put conditions on a closed
Mn to ensure that deg F : Mn � Tn must vanish.
CHAPTER
14
Harmonic Forms
1 4. 1 . The Hodge Operators
What are Maxwell's equations in a curved space-time?
14.1a. The * Operator On a (pseudo-)Riemannian manifold Mn we introduce a pointwise scalar product be tween p-forms by ( 1 4. 1 )
where, as usual, I = (i " . . . , i 1' ) , and denotes that i n the implied sum we have i, < i 2 < . . . < i p ' It is not difficult to check that if e = e" . . . , en is an orthonormal basis for tangent vectors at a point, then a ' , . . . , a n is an orthonormal basis of I -forms and also that ---r
yields an orthonormal basis for p-forms at the point for i , < . . . < i p • We now introduce a global or Hilbert space scalar product by (a P , f3 P ) : =
i
(a I' , f3 p ) voln
( 1 4.2)
whenever this makes sense; this will be the case when M is compact, or, more generally, when a or f3 has compact support. We should remark at this time that the space of smooth p-forms on a Riemannian M that satisfy (a I' , a p ) < 00 form only a pre-Hilbert space since it is not complete; a limit of square integrable smooth forms need not even be continuous ! To get a Hilbert space we must "complete" this space. We shall not be concerned here with such matters, and we shall continue to use the inaccurate description "Hilbert space." We shall even go a step further and use this denomination even in the pseudo-Riemannian case, when (,) is not even positive definite !
361
HARMONIC FORMS
362
I f a ' i s a I -form, w e may look at its contravariant version A, and to this vector We may associate the pseudo ( n - 1 ) -form iA vol" . In this way we associate to each I -fonn a pseudo ( n - I ) -form. We are now going to generalize this procedure, aSS O Ciatin g to each p-form aP a pseudo (n - p) -form *a, the (Hodge-) dual of a, as follow s . If aP = a1 dx i then where
( 14.3 )
If f is a function we have ( 14.4)
Written out in full
and where the upper indices K in a K indicate that all of the covariant indices in a have been raised by the metric tensor,
For an important special case, the O-form that is the constant function f = 1 has
* I = JjgTE ' 2 ... " dx '
/\ . . . /\
dx"
= vol"
( 1 4.5 )
Note that for a given i, < h < . . . < ill - p , there is at most one nonvanishing term in the sum on the right side of ( 1 4. 3) , namely when k , < . . . < k p is the complementary multi index to i , . . . ill - I" We then have
and
and so ( 14.6)
This shows that indeed * takes forms into pseudoforms and conversely. We have claimed that * generalizes the map a ' � i A vol" . To see this, iA voln
i A ME1 dx i
=
MA j Eg : dx k2
/\
•
•
•
/\
dX k"
iA vol" = *a '
==
( 1 4.7 )
Equation ( 1 4. 3) is frequently awkward to apply; many times it is more convenient to use directly ( 1 4.6) together with the following. Let e = (e " . . . , en ) be an orthonormal
THE HODGE OPERATORS
363
frame of vectors (allowing I e l 11 2 = - I in the case of a pseudo-Riemannian manifold) ; I I as w e have mentioned, a , for I = ( i I , . . . , i p ) , are then al s o orthonormal and a /\ . . . /\ (J " :::: ± vol" . Thus, from ( 1 4.6),
*a l
=
±a J
( 14.8)
I = (i I , . . . , i p ) . example, at the electromagnetic field in a perhaps curved space-time for Look , 4 manifold M . This will be discussed in more detail in Section 14. l c. We shall see there that the fi eld is again described in local coordinates (t , x) by the 2-form where J = (j l , . . . , ill - I' ) is complementary to
w here
and
B23 dx 2 /\ dx 3 + B3 1 dx 3 /\ dx l + B I 2 dx l /\ dx 2 Then, using the space-time metric, * F2 will again be a 2-form, and so will be of the �g2
=
form
*F
=
* (i:� 1 /\ dt) + *�g 2
=
[*l-� I ] - [*�Jl 2 /\ dt]
for some spatial I -form *�Jl 2 and some spatial 2-form *(:� I . Let us find these forms in the special case of Minkowski space, without using ( 1 4.3). * takes p-forms into pseudo (4 - p) -forms. �t12 = B l dx 2 dx 3 + B dx 3 /\dx l + B dx l /\ 3 2 dx2, is a 2-form in Minkowski space-time. Since the coordinates are orthonormal and M = I , we can probably avoid the use of ( 14.3). * (dx 2 /\ dx 3 ) has the property that
(dx 2 /\ dx 3 ) /\ * (dx 2 /\ dx 3 ) = 11 dx 2 /\ dx 3 11 2 dt /\ dx l /\ dx 2 /\ dx 3 • Since the dx'" are orthonormal and II dx'" 11 2= + I for a = 1 , 2, 3, we see that II dx 2 /\ dx 3 1 1 2 = II dx2 11 2 11 dx 3 11 2 = + I , and so *(dx 2 /\ dx 3 ) = dt /\ dX I . Likewise for the other two
terms. We then have, from Equation (3.4 1 ), =
- (Bl dx 1 + B2 dx 2 + B3 dx 3 ) /\ dt = _ (*�t1 2 ) /\ dt Note that *�jj 2 is simply the star operator in JR. 3 (which takes p-forms to (3 - p ) -forms) applied to the 2-form �jI 2 . In our older notation it is simply ( , B ) , as in Equation (3.4 1 ) ! Look now at the term l-� I /\ d t = (E l dx l + E2 dx 2 + E3 dx 3 ) /\ dt . For example * (dx 1 /\ dt) = - I I dx 1 /\ dt 11 2 dx 2 /\ dx 3 = dx 2 /\ dx 3 sinc e I I dt 11 2 = - 1 Thus * (i:� 1 /\ dt) = E l dx 2 /\ dx 3 + E2 dx 3 /\ dx 1 + E 3 dx 1 /\ dx 2 , *�il 2
.
that is,
where *<'i, 1
* (i:� l /\ dt) =
=
*I;}
i E vol 3 results from applying the star operator ofJR.3 to (:� l . This explains our use of the notation * F 2 in Section 7.2h and the use of * in Section 3.5c. This concludes our electromagnetic excursion for the moment.
364
HARMONIC FORMS
I n Problem 1 4. 1 ( 1 ) you are asked t o show that * (*a P )
=
( _ l ) p (n - p)a
if Mn is Riemannian
- ( _ l ) p(n -p)a
( 1 4.9)
if Mn is pseudo-Riemannian
It is sufficient to verify these for terms of the form a I and to assume these are ortho normal. Finally, note the following. If A is a vector and a is its associated I -form, then *a is a pseudo- en - I )-form, and if vn - 1 C Mil is a transversally oriented hypersurface, then ( 1 4. 1 0)
In particular
Iv *dj Iv (V j, N}dSIl - 1 =
for any function j, and this last integral is the "surface" integral of the normal deriva tive df/ dN over the hypersurface.
14.1h. The Codifferential Operator 8
=
d*
Exterior differentiation d : I\ P M n ---+ (p + I ) -forms; in this section we shall exhibit an operator that decreases the degree of a form by one, and, in the case of a compact manifold, serves as the pre-Hilbert space adjoint of d. We thus want an operator I\ P + l M n sends p-forms to
d*
:
such that
p- l
/\ ---+ /\ p
( 1 4. 1 1 )
Now (da P- 1 , f3 P ) = fM da P - 1 /\ *f3 P . Consider first the Riemannian case; we m ay then use the first equation in ( 1 4.9). Note then d (a
/\
*13)
/\
=
da
=
da
=
d a /\
/\
*13 + ( - l ) p- 1 a
/\
d * 13
*13 + ( _ l ) p- l (_ l ) (n - p + l ) (p- l ) a *13 + ( _ l )n(p+ l ) a
/\
*
/\
**
d * 13
* d * 13
and so
with a similar result for the non-Riemannian case. We then define whether M is compact
or not and whether or not M has a boundary d* f3P
:
=
( - l r(p+ l ) + l * ( _ l ) n (p + l ) *
d * f3 P Riemannian
d * f3 P pseudo-Riemannian
( 1 4 . 1 2)
365
THE HODGE OPERATORS
and then
(da P- 1 , fJ P ) - (a P- 1 , d * fJ P )
=
L d(a P- 1
1\
*fJ P )
( 1 4. 1 3 )
at least if a or fJ has compact support. If Mn is a closed manifold, then d * , as defined i n (1 4. 1 2 ), is the pre-Hilbert space adjoint of d. If M is a compact manifold with boundary a M , let i : aM ---+ M be inclusion. Then
(da P - 1 , fJ P ) - (a P- 1 , d * fJ P )
=
1
aM
aP- 1
1\
*fJ P
( 1 4 . 1 4)
an d then d * is again the adjoint of d if we restrict ourselves to one of two types of foOll s : those forms y that are 0 when restricted to the boundary, that is, i * y = 0, or those forms y whose dual *y is 0 when restricted to the boundary, i * * Y = o. The operator d * is called the codifferential operator. The traditional notation for d * is 8
8
:=
d*
but we shall avoid this notation since the symbol d * is more informative and we prefer to reserve
8 for the variational symbol.
We shall need a coordinate expression for the (p - I )-form d * fJP .
Theorem (1 4.15): (d * {3P ) K
=
-fJ j K jj
We shall call the negative of the right-hand side the Divergence (with a capital D) of the form fJ
although sometimes it will look more like a curl ! Note that this is the same definition as for the Divergence of a symmetric tensor in Equation ( 1 1 . I 5 ) ! We only define the Divergence of a tensor that is either symmetric or skew symmetric. we gave
PRO O F :
To show that two (p I )-forms y and p are identical we need only show given any small closed coordinate ball B (disjoint from aM if M has a boundary) then for all (p - 1 ) -forms a whose support lies in the interior of the ball, IE (a , y ) * 1 = IE (a, p) * 1, for if the volume integral of (y T p T ) vanishes for all smooth a and for each small ball, then y p = O. We shall verify ( 1 4. 1 5 ) by showing that -
-
al
is (a P- 1 , d* fJ P ) * 1 is (a P- 1 , - DivfJ P ) * =
-
1
We may consider the new manifold-with-boundary B instead of M . For this manifold the preceding integrals are inner products, and we must show, since a vanishes on the boundary of the ball,
(a P - 1 , d * fJ P )
=
(a P- 1 , -Div fJ P )
366
HARMONIC FORMS
1 1 .2 (1) (aP - 1 , d* f3P) (daP - 1, f3P) is (da, (3 ) * 1 is (da)l f3 ' * is 8r a fW f3 ' * 1 is (8r a li f3 ' ) fj * - is a li8r f3;j * 1
Using Problem
=
=
1
=
=
1
=
But
8r f3 ' f3 j K
(why?)
=
and so
(aP - 1 , d*f3P) = is (alif3 j K) /j * l - is a li f3 j K /j * l In the first integral, C j alif3 j K [(p - 1)W 1 aKf3 j K are the components of a contravariant vector C, and then the integrand is the divergence of this vector. Ja B ( C , N)d B ut JB div C * 1 since C vanishes on B . Thus (a P - 1 , d* f3P) j� alif3 j K /j * I is (a, -Div(3 ) * 1 :=
=
S
=
=
as desired.
a
0,
=
=
-
D
14.1c. Maxwell's Equations in Curved Space-Time M4 We shall assume that the electromagnetic field is again described by an electromagnetic 2-form F 2 . In any local coordinates (t = x o , x) we may decompose F 2 into a part that contains dt and a part without dt; thus F 2 defines an electric I -form � ) and a magnetic 2-form �Jl 2 through
but of course this decomposition depends on the coordinates used. We postulate that for any bounding 2-cycle Z 2 = a u 3 in space-time M 4 we have
/'
Ja u
F
2
=
( 1 4 . 1 6)
0
If F is continuously differentiable, we conclude that Ju
dF
= O. Since U can be chosen to be an arbitrarily small hypersurface with arbitrarily chosen normal, we see that we must then have
d
d
This is the first set of Maxwell equations. If we write, as usual, = d + t 1\ a/at, F = 0 yields the usual Maxwell equations (3.39) and (3 .40), together with their primed differential versions. Note that the operator is i ndependent of the metric of space-time. We postulate that there is a current pseudo-3-form, with associated decomposition
d
d
",3
c'
_ -
a 3 - ·"2
J
1\
dt
367
THE HODGE OPERATORS
Since the notion of the charge contained in a region is independent of the metric, S3 is assumed given independent of the metric. Of course, y, 3 can be written in the form
bu t the current 4-vector J will depend on the metric ! It is for this reason that S 3 i s more basic than J . We then postulate that for any 3-cycle Z 3 , bounding or not, we have
h j',3
=
0
( 1 4. 17)
If one applies this to the boundary of a solid space-time cylinder
Z
=
one sees that this is conservation of charge (this i s Problem 1 4. 1 (4». We now postulate that
f *F
Jau
=
4n
f y,3
a { V3
x
[0, T ] }
( 1 4. 1 8)
Ju
for all 3-chains U . Note that this is compatible with ( 1 4. 1 7). This is the second set of M axwell equations. When y, is smooth we see from the same argument as used after (14. 1 6) that y,3 is closed, d;!,3 = O. Since the periods of ;!, 3 vanish, we conclude from de Rham that y, 3 is in fact exact, and postulate ( 1 4. 1 8) says essentially that * F 2 is a "potential" for ;!,3 ! ( 1 4. 1 9) Since
*F i s a 2-form we may define pseudoforms *(� I *F
=
_ ( *Hl 2 )
and *fJ} 2 by
1\ dt + *(�I
( 1 4.20)
It is no longer true that *(� I and *f1l2 are the Hodge duals (using the 3-space metric gafJ of the spatial section t = constant), g OfJ =I- 0, *fH 2 may involve &. as well as fil l
of the forms
i:� 1
and fJ} 2 ! If, for example,
In the smooth case the second set of Maxwell ' s equations ( 14. 1 9) are exactly as in Minkowski space, that is, (3 .42' ) and (3 . 43 ' ) . Maxwell 's equations in curved space are exactly as in fiat space, once we accept * F as defining the fields *f1l 2 and *i:� I .
14.1d. The Hilbert Lagrangian The Hilbert action for Einstein's theory is essentially iM R * 1 . Although the curvature matrix e is a matrix of 2-forms, we haven ' t expressed either the Ricci tensor (which is symmetric) or the scalar curvature in terms of forms. Still it is possible to write the action in terms of forms ; although the expression is awkward, it does occur in physics papers and the reader should be aware of it. We shall be very brief. e a b = Ra b(r < s)dx r 1\ dxs is a matrix of 2-forms. Then * e a b is defined to be the matrix obtained by taking the * of each of the 2-forms, that is, * does not affect the indices a h. Then
* e ll h
=
cd
I g 1 1 / 2 R a h E (c
368
HARMONIC FORMS
and
dX a
1\
d X b 1\ * eah
Rah cd I g 1 1 / 2 E( c
=
=
=
=
=
Thus
=
P roblems
1 4. 1 (1 ) Verify ( 1 4.9) .
fJ P ( D ivfJ P) K {3jK j 1 9 1-1/2alaxj(1911/2fJjK) / Note that if f and 9 are functions then v 2 f - d* d f and if M is compact ( f, v 2 g) = JM fV 2 9 * 1 . Apply Equation ( 1 4. 1 4) in the case when Mn is a
1 4. 1 (2) Show that for any p-form
=
1 4. 1 (3)
=
=
compact manifold with bou ndary to obtain Green 's theorem
r ( fV2 g _ gv2 f) * 1 JarM f * dg - g * d f
JM
=
1 4.1 (4) S how that ( 1 4. 1 7) does imply conservation of charge.
14.2. Harmonic Forms
Among all closed forms with a given set of periods, which one has the smallest global norm?
14.2a. The Laplace Operator on Forms In ]R1l with cartesian coordinates, the Laplacian of a function f is the familiar V 2 f = Lea2 fjax i axi ) . We have given two equivalent invariant expressions for V2 on a Rie mannian manifold in Equations e2. 89) and e 1 1 . 29) . The Laplacian of a p-form field is a more complicated matter. Consider a vector field A . In JR.n with cartesian coordinates, one could define V2 A to be the vector field whose components eV2A)i = L j ea2Aijaxjax j ) are simply the Laplacians of the components of A, considered as functions. In JR. 3 this can be expressed in the usual form found in physics books,
V2A
=
grad div A
-
curl curl A
( 1 4. 2 1 )
We can write this expression in intrinsic form i f w e consider the covector a I associated to A, instead of A itself. Note first that from Equation ( 1 4. 1 5 )
d*a l
=
- div A
369
HARMONIC FORMS
and so the covariant version of the first term in ( 1 4.2 1 ) is -dd * a . Furthermore, da ' 2 i s the 2-form version of curl A. For any 2-form /3 = i (B) vol we have, from ( 1 4. 1 2), 3 2 d* {32 == ( _ 1 ) (3)( ) + 1 * d * /3 2 = *d * /3 2 . */3 is the I -form version of B and so d * /3 2 is the 2-form version of curl B and *d * /3 2 is the I -form version of curl B. Thus -d * da 1 is the I -form version of -curl curl A. Finally then ( 1 4.2 1 ) has as covariant version
V2a l
=
We shall define the Laplace operator that is,
�
1\ -+ 1\
d * d)a '
� on p-form by the negative of the preceding,
p
p
:
+
- (dd *
by �
:=
dd * + d * d
( 14.22)
-�.
Occasionally we shall write V 2 : = Note that from d 2 = 0 and * * = ± 1 , we have
d *d*
=
� =
± (*d*) (*d*) (d + d * ) 2
=
0
and so ( 1 4.23)
In Problem 14.2( 1 ) you are asked to show the following in ffi. 3 , using brief explanations as we did in deriving part 6 in the following
� in
ffi. 3
1. d* f a
= o. 2. d * a l = - div A. 3. d* f3 2 = d* iB vol 3 *icurlB vol 3 is the I -form version of curl B. 4. d * y 3 = d * (*g o ) * dg is the 2-form version of - grad g . 5 . � o = _ \7 2 f a f . 6. �ex l is the I -form version of curl curl A grad div A . 7 . �f3 2 = is the 2-form version o f curl curl B - grad div B. 8 . � (* f a ) - * ( \7 2 f ) . =
=
-
-
=
14.2h. The Laplacian of a I-Form
Let ex l = a i dx i be a I -form on a Riemannian Mil . We shall compute a coordinate expression for = (dd * + d * d)a . First
�a
da
==
2 )ai a j - aj ai )dxi /\ dx j i <j
=:
=
2 )a j li - ai /j )dx i /\ dx j i <j
L Cij dx i /\ dx j
i <j I ( d* C ) j - -Cji l i _ - -a /j i I i wh ere we have put
i + a lj i
370
HARMONIC FORMS
Thus
(d*da ) j -- aj I i I i -
Also
+
i a lji
d*a -air and so d(d*a) - a� i d j , that is, (dd*a) j a/ j Thus =
j
=
x
=
-
i
'
By Ricci's identity ( 1 1 .23)
(
We conclude
ua ) j - -aj I i I i + ak Rkii j - -aj I i I i + a k Rkj A
_
( 1 4 . 24)
_
( 1 4 25 ) .
The first term in ( 1 4.24) looks, at first glance, as if we are taking the negative of the usual Laplacian of the componentfu nction a , but this is not so since a l i = ai a j -akrr, and this connection coefficient would not occur in the covariant derivative of a functio�. The first term in ( 1 4.25) is sometimes called a "rough" Laplacian, written It differs from the Laplacian (defined first by Kodaira and independently by Bidal an d de Rham) by the second term in ( 14.25), which does not involve any derivatives of a !
j
j
VVa.
�a
-(�a) j -('V'Va) j + ak R k j
(1 4 .26)
=
( 1 4.25) and ( 1 4.26) are called Weizenbock formulae.
14.2c. Harmonic Forms on Closed Manifolds
Let Mn be a compact Riemannian (rather than pseudo-Riemannian) manifold. Then the global inner product (,) is positive definite, for
1 a 1\
1 (a, 1 and at the pole of a geodesic coordinate system (a, a) I: (alY ' Thus (a, a) and vanishes only if a vanishes identically. We say that a form aP is harmonic if �a O. For a function (i.e. , O-form) this
(aP,
f3 P ) =
f3 ) *
*f3 =
::: 0,
=
=
reduces to the usual notion. Let Mn be a closed manifold. If we again denote the formal adjoint of an operator A on forms by A then since = we see that is formally self-adjoint, = Furthermore,
*, � (d +d*) (d +d*), � � * �. (�aP, aP) (dd*a + d*da, a) (d*a, d* a) (da, da) da + I d*a +
=
=
which is :::: 0 in our Riemannian case. Thus
�a
=
0
iff
da
=
0
and
= 11
d*a
=
0
Harmonic forms on a closed manifold are both closed and coclosed!
11 2
11 2
( 1 4.27)
371
HARMONIC FORMS
This is far different from the situation in ]Rn . For example, a closed O-form is simply constan t fun ction, yet harmonic functions in ]R" need not be constant; the real part of
any
complex analytic function in the plane is harmonic ! The Lapl ace operator � : I\ P -+ I\ P is an elliptic operator on a Riemannian manifold (for the noti on of ellipticity and for the proof of Hodge's theorem later, see [W, chap. 6]); the m ain in gredient is that the metric tensor is positive definite. In Minkowski space, how ever, the Laplacian of a function becomes the d' Alembertian a
"
where "172 is the spatial Laplacian;
a2 f �f = -2 - V 2 f at
�
in this case is the wave-operator and is hyperbolic. esults in elliptic operator theory are needed for the following fundamental t r cul ffi Di
result:
Hodge's Theorem (14.28): Let M" be a closed Riemannian manifold. Then the vector space of harmonic p-forms :fCP is finite-dimensional, has a
=
{
h
A I dh
E
=
0
=
d*h
and Poisson 's equation �a P
=
}
pP
solution a iff p is orthogonal to :lfP (pP , hP)
=
0
for all h P
E :1[1'
The finite dimensionality of :i{P is a deep result on elliptic operators on closed manifolds. On th e other hand, it is easy to see the necessity of the condition on p in order that there be a solution to Poisson's equation; if �a = p, then for h E :fCP,
(p , h )
=
(�a, h)
=
(a, � * h) = (a, �h)
=
0
The deep part is showing the sufficiency of this condition. Note also that in the case p = 0, that is, when we are dealing with functions, the harmonic function h is then a constant, and the condition on p is simply that
L p vol"
=
(p , 1 )
=0
that is, p must have mean value 0 on M. This is of course necessary since
L �a o voln = L div (grad a n ) vol" -
by the
=
0
divergence theorem. Suppose now that fJ P is an arbitrary p-form on the closed M" . Let h I , h 2 ' . . . , h r be an orthonormal basis for the harmonic forms :fCP . Then
372
HARMONIC FORMS
:J(P and so, by Hodge's theorem, we can solve D.aP fJl' hP for aP. I n other words, for any f3P o n M " w e can write
i s orthogonal t o
=
-
( 1 4.29)
f3 on the closed M" can be written as the sum ofan exactform d(d*a) plus a coexactform d*(da) plus a harmonic form. Hence P/\ d p/\- I + d* p/\+l Thus, any plorm
+ :J(
=
p
( 1 4 .30)
Note further that the three subspaces are mutually orthogonal
(dy, d*/J.,) (dy, h) (d*f.L, h ) =
=
=
0
( 1 4. 30) is called the Hodge decomposition of /",P . Note that the decomposition ( 1 4.30) is unique. If we write = +h = then orthogonality = 0, = 0, and h h = O. Note also that we are not saying, for example, that y is unique, for clearly we c an add to any closed (p I ) -form; we are only saying that is a unique summand. At first glance it might appear that ( 1 4.30) is a triviality, for we can see immediately that xl' is the orthogonal complement in /'/ to the direct sum of the exact and coexact forms ; if for some p- form = 0 and = 0 for all and then indeed = 0 = h and so is harmonic and thus = :-1(1' . However, I\P is an infinite-dimensional space, and in infinite dimensions it is not necessarily true that if is a subspace then 1- is the entire space ! It is true that if A is a closed subspace of a Hilbert space, then 1- is the entire space. Thus to get the decomposition ( 1 4.30) one might first complete the pre-Hilbert space to a Hilbert space, say the square integrable forms on Mn ; we would have to consider forms that are not even continuous, and for such forms is not defined ! In any case would not be a closed subspace. All these difficulties can be overcome by invoking elliptic operator theory, and we refer the reader again to [W] for this difficult material. In the case of a closed 3-manifold we have = that is,
dy' + d*f.L' +h', y p- I d* h A
givesdy-dy'
-
h, (d y, h) h A+A A +A
d
d
B
=
f3 dy + d*f.L d*f.L-d*f.L' ' dy -
(d* f.L, h) y f.L, [d /\ p- I +d* /\ 1'+ 1 ]1/\ P [d /\ p - I +d* /\P + I ]
f31 d¢o + d* f.L2 + h I, grad ¢ + curl M + H
that is, a smooth vector field can be written as the sum of a gradient, a curl, and a vector field that has both vanishing curl and divergence. Thus it is true that any vector field B can be written as the sum of a vector field with vanishing curl and a vector field with vanishing divergence. This version is also true in the noncompact JR; 3 , at least when the growth of B at infinity is controlled; this is the classical Helmholtz decomposition, which is so useful in vector analysis.
14.2d. Harmonic Forms and De Rham's Theorem We now have the following picture illustrating the orthogonal Hodge decompos iti on on a closed manifold.
373
HARMONIC FORMS
H I'
,1------ co-closed
A----'--"--.c-:- fl
, .. .. I .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .:- ....
#
----
----
.
Figure 1 4.1 Any p-form fJ may be written in the form fJP = daP- 1 + d* y p+ 1 + hP where h is harmonic. In particular, since the decomposition is orthogonal,
Corollary (14.31): If fJP is closed, dfJ P
=
0, on a closed manifold Mn, then
fJP = da p - I + hP where h P is harmonic. Now fJ and fJ - da are in the same de Rham class. Thus
Corollary (14.32): In each de Rham class [fJ ] there is a unique harmonic repre sentative h (fJ) . Thus there exists a unique harmonic p-form with b p prescribed periods on a homology basis for the real p-cycles on Mn. Riemann was aware of this in the case of a closed surface. A "proof" goes along the fol lowing lines. Assume that one has a closed p-form fJ P on a closed manifold Mn . (Closed I -forms on an M 2 with prescribed periods are easy to construct, as we did in Section l 3 .4b. ) The I -parameter family of forms fJP (E) := fJP + Eda p- I are clo sed, with the same periods, for all (p - I ) -forms a . This yields a variation of fJ with 8fJ = da . Suppose that fJ is the closed form with the prescribed periods whose norm is a m inimum. Dirichlet 's principle presumed that such a minimum norm element had to exist. Look then at the first variation as we vary a 0 = 8 (fJ , fJ ) = 2(8fJ, fJ)
= 2(da, fJ)
=
2(a, d * fJ)
Since this holds for all a we conclude that fJ is not only closed, it is coclosed, d * fJ and thus harmonic !
=
0,
374
HARMONIC FORMS
It was pointed out b y Weierstrass that Dirichlet's principle was not always re liable and thus the indicated proof is defective. Note that the (difficult) Hodge decomposition justifies the norm claim since II f3 112:::: II da 1 1 2 I I h 11 2 shows that in the de Rham class [,8] , the harmonic repreSenta tive h •
+
has the smallest norm!
14.2e. Bochner's Theorem Let us say that M" has positive Ricci curvature if the Ricci tensor is positive definite
,
This is a weaker condition than positive (sectional) curvature since this quadratic fonn represents a sum of sectional curvatures (see ( 1 1 .67)).
Bochner's Theorem (14.33): If the closed Riemannian Mn has positive Ricci curvature, then a harmonic i -form must vanish identically, and thus M has first Betti number b l = O. Let us compute, with Bochner, the Laplacian of the square of the point wise length (h , h ) = h i h i of any harmonic I -form h . First,
PROOF:
[grad (h , h ) L = 2h , /j h ' and so
. 1 \7 2 2 (h , h ) - (h , /J h i ) /j - h i /J. /j h , + h , /j h i/j j = h /j /j h ' + h i /j h i / B y ( 1 4.25) we have, since f.. h = 0, h, /j /j = h k Rk i , and thus _
1
\7 2 2 (h , h )
=
_
Ric (h , h) + h ' /jhl/J ::: Ric(h , h) ::: 0
But then 0 = JM \7 2 ( 1 /2) (h , h ) * 1 ::: definite, that h = O. 0
JM Ric(h , h) * I shows, since Ric is positive
Bochner's theorem should be compared to Synge's corollary ( 1 2. 1 0) . Before doi ng so, we need a general observation about closed curves. A closed (oriented) curve C on Mn represents an element of the first homology group HI (M ; G) for any coefficient group G . If C is contractible to a point, then in the process of shrinking, the curve will sweep out a surface, of which it is the boundary. In other words, if a closed curve can be contracted to a point then this curve bounds, that is, trivial as a I -cycle. (In particular, if M is simply connected, then HI (M; G) == 0. ) The converse is not true.
375
B OU N D A RY V A L U E S , R E L A T I V E H O M O L O G Y , A N D M O R S E T H E O R Y
c
o Figure 1 4.2
The edge C of the punctured torus M is clearly the boundary of the surface M, and so is hom ologically trivial, but it seems rather clear (and can be proved) that C cannot be shrunk to a point because of the presence of the "hole." As far as Betti numbers are concerned then, Bochner's theorem is stronger than Sy nge 's corollary since positive Ricci curvature is weaker than positive sectional cur vature, and also we do not require even dimensionality nor orientability, but it should be kept in mind that simple connectivity is a stronger notion than hI = O. P roblems
14.2(1 )
Derive all those equations ( 1 ) through (8) that have not been discussed p revi ously.
1 4.2(2) 1 4.2(3)
Show that
Ll
commutes with d, d * , and
*.
Show that if Mn is closed and orientable then bp bn - p. This is a special case of Poincare d ual ity. W hy do we need orientabil ity? I l l ustrate with bo for the 2-torus and the Klein bottle. =
14.3. Boundary Values, Relative Homology, and Morse Theory
What does topology have to do with the existence and uniqueness of physical fields? The
prime example of a manifold with boundary is the case of a bounded region in
]R3 with smooth boundary. If a fluid fills such a domain, with smooth walls forming
the boundary,
then the velocity vector field v is tangent to the boundary. If the flow is incompressible, then the velocity field has divergence O. If further the flow is irrotational, then the velocity has curl 0 and the resulting velocity I -form field l! is harmonic. We are interested in the existence of such fields and we shall find that with some type of prescribed topological restriction the solution becomes unique. Note that in a compact manifold with boundary, Equation ( 14. 1 4) shows that the operators and are not necessarily adjoints, and it is no longer true that = 0
d d*
b.a
376
HARMONIC FORMS
�
i ff da = 0 = d * a . Furthermore, is no longer self-adjoint. For physical probl ems involving forms we shall reserve the term harmonic field for forms that satisfy
da = 0 = d * a
Thus a harmonic O-field is constant, whereas a harmonic function, that is O-form of , course, need not be.
14.3a. Tangential and Normal Differential Forms Let Mil be a compact Riemannian manifold with boundary. A form aP on M is said to be normal to a M, or simply normal, provided the restriction i * a of a to the boundary vanishes, i * a = 0
where i : a M � M is the inclusion map. Recall that this simply means that a (v , . . . , w) = 0 when v, . . . , w are all tangent to a M . If we suppose that a M is locally defi ned in the coordinate system x I . . . , x n by putting x n = 0, then ,
aP is normal
iff a P = dx"
/\ y p
-I
for some form y . For example, a I -form a l i s normal provided a l = an (x ) dx n (no sum ! ) at points of a M . If T is tangent to a M, then 0 = a (T) = (a, T) shows that a l is normal
iff a is normal to a M
where a i s the contravariant version of a l . If, however, /3"- 1 i s an ( n - I )-form, /3 n - 1 = i B voln , then /3 is normal provided /3 (T2 , . . . , Tn ) = vol/ (B , T2 , . . . , Til ) = 0 for tangent T; ; and so
/3 n -1
=
i B voln is normal
iff B is tangent to a M
A form aP i s said to b e tangent to a M , or simply tangent, provided * a i s normal, = O. Thus
i* * a
a I is tangent
iff a is tangent to a M
while
/3 n- 1
=
i B voln is tangent
iff B is normal to a M
Note that from the remark following ( 1 4. 1 4), d * is the adjoint of d if we restrict ourselves either to tangential or to normal forms ! I n the following we shall quote, without proofs, the versions of Hodge's theorem that have immediate applications to physical problems. My principal guide for the ap plications has been the mimeographed NYU notes [B , F, G) by A. B lank, K. Friedrichs, and H. Grad of 1 957. For the (difficult) mathematics of harmonic forms on manifolds with boundary, the reader may consult [D, S) and [Fk) .
BOUNDARY VALUES, RELATIVE HOMOLOGY, AND MORSE THEORY
377
14.3b. Hodge's Theorem for Tangential Forms Theorem (14.34): Let M n be a compact manifold with boundary. Let Z I , . . . , Zbp be a basisfor the pth homology vector space HI' (M; JR). Then there exists a unique
tangent harmonic p-form field a "
da " = d * a " = 0 with prescribed periods
Jz a l' on the given basis.
In other words, Hodge's original theorem holds for tangential harmonic fields in the ca se of a manifold with boundary ! Example 1 : Let v be the velocity field for a steady incompressible, irrotational fluid flow inside a closed surface V 2 of genus g. As we have seen, v i is harmonic, d v = d * v = 0, and v is tangent to a M .
Figure 1 4.3 We shall illustrate the case for genus 2. M 3 is the solid "pretzel," and a M is the surface of genus 2. It should be rather clear that a homology basis for HI ( M ; lR) is given by the two indicated I -cycles circling the "holes." The period of v i on a I -cycle z, Jz v i , is called the circulation of v around z . Thus Hodge's theorem yields the following corollary, known to W. Thomson (Lord Kelvin).
Corollary (14.35): There exists a unique incompressible irrotationalflow inside a suiface of genus g with prescribed circulations around the g holes. In particular, if all circulations vanish, then the fluid must be at rest ! This is the only possibility in the case of a spherical surface since the solid ball has first Betti number O.
Example 2: Let M 3 be the region inside a closed conducting surface Vo and closed conducting surfaces V and V .
I
2
outside
378
HARMONIC FORMS
Figure 1 4.4 We have drawn the case when Vo is a large ellipsoid, VI is an interior 2-sphere , and
V2 is an interior 2-torus . Consider an electrostatic problem in which there are no charges
inside M 3 ; of course there may be charges interior to VI and/or V2 or exterior to Vo. 4JT (J 3 = ° and d * *G� = *d * *6 == Then the electric field inside M3 satisfies d*�� 1 2 0, and so *�, is a harmonic 2-form in M 3 . Since a tangential *d{� = * [ - a �J3 / a t ] component of E would give rise to currents, that is, moving charges, in a conductor, the natural boundary condition for electrostatics is that E be normal to conducting surface s. Thus *�; is a tangent harmonic 2 -form field in M 3 . Note that a M 3 = Vo + VI + V2 , and thus a plausible (and correct) basis for H2 (M3; lR) i s, for example, VI and V2 . Thus there exists a unique electric field in M 3 with prescribed periods J J *c; over VI and V2 • But the integral of *�� over Vi is 4JT Qi , where Qi is the total charge inside Vi . =
=
Corollary (14.36): There exists a unique static electric field E in M 3 with preas signed charges in the cavities VI and V2 . The field is thus independent not only of charges outside Vo ( "shielding "), but also of the exact placement of the charges in VI and in V2 • We should mention that Theorem ( 1 4.34) is a special case of a more general result. First recall that to say that aP is "tangent" is to say that the restriction i * ( * a) of *0' to the boundary vanishes. More generally, we could ask for a harmonic field a P that has prescribed periods and such that i * (* a) is a prescribed form y " - P on a M . The spec ial case y = ° would make a a tangent form. We must put some restrictions on the form y for the following reason. On a M we have d y = di* * a = i *d * a = 0, since a is coclosed. Hence y is closed. Furthermore, y is only defined on a M , but suppose that Z,,- P is a cycle on a M that bounds in M, that is, i z = a c , for some (n - p + I )-chain c * on M . Then since the integral of y over z is the same as the integral of * a over z, this integral must vanish, * a being closed on M . The following notion is due to Tucker.
A.
Definition (14.37): An admissible boundary value form y r on a M is a closed form on a M whose integral vanishes on every cycle Z r on a M that bounds on M.
B O U N D A RY V A L U E S , RELATIVE HOMOLOGY , A N D M O R S E THEORY
379
The generalization of ( 1 4.34) is as follows. (For more along these lines see [D, S ] . )
Theorem (14.38): There exists a unique harmonic field a P on M with prescribed periods and whose dual * a restricts on a M to a prescribed admissible boundary value form yn - p • The uniqueness of a is simple (and was known to Lord Kelvin in the case p = 1 ). P R O O F O F U N I Q U E N E S S : Let a P be a solution and suppose f3 P is another with
the same periods and whose dual * f3 has the same boundary values i * * f3 = y . Then fL : = a - f3 is a tangent harmonic field with 0 periods. S ince dfL P = 0, fLP = dvp- I for some v (this is elementary if p = 1 ; otherwise it requires de Rham's th eorem) . We wish to show that dv = O. But
(dv, dv)
=
r dv
1M
/\
*dv
=
1aM (v
/\
*dv) ±
rv
1M
/\
d * dv
Since fL = dv is tangent, * dv is normal and the boundary integral vanishes. Also d * dv = d * fL = 0 since fL is harmonic. 0
14.3c. Relative Homology Groups The topological "cycles" that we have been involved with so far are called absolute cycles. Given a compact manifold Mn perhaps with boundary we can define a
relative p-cycle (mod a M) be a p-chain on M whose boundary, if there is one, lies on a M . Of course every (absolute) cycle is also a relative cycle.
to
Figure 1 4.5 In
Figure 14.5 the curves C 1 , C2 , and C3 are all relative I -cycles (mod aM = + V2 ) . We shall systematically disregard any chain that lies on a M . That is
Vo + VI
380
HARMONIC FORMS
why w e may think o f a relative cycle a s a cycle; w e may disregard its boundary s ince it lies on a M . We shall say that two relative p -cycles c and c ' are homologous (mod a M) provided they differ by a true boundary plus, perhaps, a p-chain that lies wholly on a M ; in other words, a relative boundary is an absolute boundary plus any chain on a M where
vp
C
aM
Figure 1 4.6 In Figure 1 4. 6 we have drawn three more curves F1 , F2 , and F3 , all lying on aM, and also an oriented 2-chain W 2 • Clearly aw = -C1 + + C3 + F2 + C2 F3 • But the F curves all lie on a M , and so we may say
FI
+
We could then say that C3 is homologous to C 1 - C2 (mod a M)
Thus only C 1 and C2 are independent relative cycles. (Of course we could have used C1 and C3 , say.) Are there any more?
Figure 1 4.7
381
BOUNDARY VALUES, RELATIVE HOMOLOGY, AND MORSE THEORY
consi der the absolute I -cycle z that threads through the toroidal hole. It is an absolute cycle of M that does not bound in M. However, as a relative I -cycle it is trivial, that i s, it bounds, since it is easily deformed i n M to lie on the torus V2 c a M . It can , i n fact, b e shown that CI and C2 form a basis for the relative homology group, HI ( M , a M ; ffi.) , defined to be the relative cycles modulo the relative boundaries HI ( M , a M ; ffi.)
=
ffi.C1 + ffi.C2
14.3d. Hodge's Theorem for Normal Forms
Theorem (14.39): Let Mil be a compact manifold with boundary. Let C I , be a basis for the relative p-cycles of M (mod a M) Then there exists a unique normal harmonic p- form
.
.
.
, c,.
aP with prescribed periods
c (mod a M ) , that is, i f c ' - c = a w p+ 1 + u " , where u lies on a M, Note that if c ' then if is closed and normal
aP
since
a" = 0
'"
w hen
j a - ja = l a = o
a is restricted to c'
C
aM!
when a Ci is replaced by a homologous C; .
II
Thus the indicated periods do not change
Example 2' : In Example 2 earlier, consider the electric field I -form t:; 1 for the elec trostatic field. It is a harmonic normal I -fo rm on M 3 . Thus we may prescribe the line integrals fc , �� l and fc, �� I . This means that instead of the charges in VI and V2 , the electric field in M 3 is uniquely determined equivalently by prescribing the electrostatic potential differences between the "inside" and the "outside" conductors ! Example 1 ' : In Example 1, we may consider the velocity vector v as defining a 2-form f3 2 = iv vol 3 . Thi s is then a normal harmonic 2-form on M 3 .
Figure 1 4.8
382
HARMONIC FORMS
I t should b e "clear" that a basis for H2 (M, a M ; IR ) is given, say, b y the two dis W I and W . Thus the harmonic normal fJ2 is determi ned by prescribing the fiU id fiu.x 2 1,Wi fJ 2 1,Wi v · dS, i = I , 2, rather than the two circulations.
:
=
14.3e. Morse's Theory of Critical Points We give here another application of relative homology groups. We shall be very brief referring the reader to Milnor's book [M] and Bott's expository paper [Bo ] for details and applications. Bott's generalization of the Morse theory of this section will be sketched in Section E. b of Appendix E.
mo�
z
5
4
3 2
"/----- y x
Figure 1 4.9 We have indicated here the height function f = z on a bumpy torus. The critical points are at levels 0 (minimum) ; 1 , 2, and 3 (saddles); and 4 and 5 (maxima). For any manifold M il with smooth real-valued function f, let us put
Ma : = { x M;; : = {x
E
E
M l f (x ) :::: a }
M l f (x ) < a }
We define a value a of the function f as homotopically critical if some relative homology group Hi (Ma , M;; ) is nonzero.
(For simplicity we shall use the real numbers IR for coefficient group, but any coefficient field can be used.) We claim that the homotopic ally critical values in our example are exactly the critical values in the sense of Section l .3d. Thus in this example the c ritical values are precisely the levels at which new relative cycles appear as we move "up" the manifold from the minimum to the maximum.
BOUN D A R Y V A L U E S , R E L A T I V E HOMOLOGY, A N D MORSE THEORY
383
z = 4 has Hz (M4 , M;; ) = lR and bited a 2-disc e at the critical point that is a generator for this homology xhi e ve ha 2 we . shall "prove" this later, but it should be plausible since any effort to slide We group tirely into the lower region M;; will require the boundary of the disc (which en thi s disc lies below z = 4) at some time to pass through the critical point; that is, the boundary wi ll have to leave the lower region at some time. It should be clear that any I -disc or O-di sc (point) on the level M4 can be pushed away from the critical point into the lower reg ion , so that Hi (M4 , M;; ) = 0 for i :j:. 2. At a non critical level b, say z = 2 . 5 , it is "clear" that any chain on Mb can be pushed Mb by a deformation along the negative gradient lines, similar to the Morse o n t i n of Section 2. l e. Thus Hi ( M2.S , M2.s ) = O. In fact, if the regions Ma are rmatio defo ed compact, and if there are no critical points Xo with d + E ::: I (xo ) ::: c E , m all as su i b then t can e shown that a modified Morse deformation (which does not move points x w ith I (x) ::: c E) can deform Md diffeomorphically into Me · At the level z = 3 , it is again "clear" that the part of any chain away from the saddle point c an be pushed down by following the negative gradient lines, but the critical point itself remains fixed. There is no continuous way to push the entire indicated I -disc e ] below level z = 3; HI (M3 , M; ) = lR with generator e] and the value 3 is again homotopically critical. We have also indicated the remaining disc generators at the other homotopically critical levels. At the minimum we have a O-disc (point) since Mo is empty. We have verified our claim. Note that the height function on the I -dimensional manifold pictured In our torus example, the relative maximum at level
-
-
z
--t----"-_=---
x
Figure 1 4. 1 0 a critical point at z = 0 , a n inflection point, but i t is clear that this does not yield a homotopically critical value since any chain on z ::: 0 can be slid below z = O. In a sense this critical point is inessential since a slight change in the function, say by tilting the z axis very slightly (in the "correct" direction), will remove the critical point. In our toral example all the ordinary critical values are homotopically critical, and vice versa, and in fact as we shall see, this is true whenever the critical points are nondegenerate in the sense of having nonsingular Hessian matrices of second partial derivatives Hij = ( a 2 fl ax i ax j ) , has
384
HARMONIC FORMS
I n this nondegenerate case w e can, with Morse, write down the dimension o f the nontrivial relative cycle at the critical point as follows. Since the Hessian is nonsi ngular there is a maximal subspace of the tangent space at the critical point on which H i� negative definite, Hij Vi v j :::: O. In terms of a Riemannian metric we are looking at the sum of the eigenspaces corresponding to the negative eigenvalues of H i j g i k Hkj • The dimension of the resulting subspace is called the (Morse-) index of the critical point, A : = number of negative eigenvalues (counted with multiplicity) and represents crudel y the dimension of the space of directions, at the critical point, in which the fun ction is decreasing. Then the relative cycle is the A-cell e),. starting out tangent to the sub space. (We shall indicate in our next paragraph why e),. does not bound as a relative cyc le.) For example, for the critical point at level 4, we can introduce new local coordinates x , y (with origin at the critical point) on the torus such that f = z = 4 - x 2 i+ higher order, and so the Hessian is negative definite on the entire tangent space to T2 at the critical point, the index is A = 2, and the disc x 2 + i < E 2 is the requ ired generator for E sufficiently small. For the critical point at level 3, in local coordin ate s 3 - x 2 + y 2 + higher order, the Hessian has the new x axis for negative f = z eigenspace, the index is A I , and x 2 < E 2 is the generating I -disc. Let us indicate why, for example, the relative I -cycle e , at level f = I is not trivial. First note that near the critical point f I x 2 + y 2 + higher order. The Morse lemma [M, p. 6] states that near a nondegenerate critical point, one may always introduce coordinates so that f becomes exactly this form with the higher order terms removed; thus in new coordinates, which we shall again call x , y , f is exactly =
-
=
=
=
-
f (x , y) = I - x 2 + l Look then at g (x , y) : = f (x , y) - I = -x 2 + y 2 . We are interested in relative cycles of the region f :::: I mod f < 1 . Away from the critical point x 0 y any chain on f :::: I can be pushed down into f < 0, and so discarded. We are then only interested in f :::: 0 near the critical point. In terms of the new coordinates we may deal with relative cycles on g = y 2 - x 2 :::: 0, that is, the shaded region in Figure 1 4. 1 1 . =
\ ---\-" )==--c- /---------
=
�:::::---------r---- y eo
x
x
----�----
y Figure 1 4. 1 1
I
BOUNDARY VALUES, RELATIVE HOMOLOGY, AND MORSE THEORY
385
Now any chain on this shaded region y 2 :": x 2 can be deformed to lie on the x axis, with no point (x , y) with g < 0 ever leaving g < O. Thus we are reduced to chains on the segment of the x axis with I x I :": 1 modulo x #- O. But x #- O on this segment can b e pu shed into the boundary x = ± 1 . Thus we are interested in the relative cycles of the segment I x I :": 1 modulo the boundary points. In the case of a critical point of index A we are interested in the relative homology of a closed A-disc B;" modulo its boun dary (A - I )-sphere S;"- I . This rather clearly (as we shall see in Problem 22.3 (3)) has only one nontrivial generator, B\ H;,. (B\ S;"- I ) = IR B;,. . In our toral case the only nontrivial generator of relative homology at the level f = I is the indicated I -cell e I , as claimed. The fact that the nondegenerate critical points are homotopic ally critical, and so have topological significance, allowed Morse to give relations between the number of critical points on M and the Betti numbers of M. Briefly we can proceed as follows. Introduce the Ath Morse type number
m;,.
: = number of critical points of index A
For bookkeeping purposes only we form the formal polynomial in a variable t with the
type numbers as coefficients, the
Morse polynomial �)n(t)
We also have the Betti numbers
nomial
b;,.
=
II
:=
L m;,.t;" ;"=0
dim. H;,. (M; IR) and the formal Poincare poly
P (t)
:=
Il
L b;,.t;" ;"=0
Morse's Theorem (14.40): Let Mil be a closed manifold and f : M -+ IR a smooth function with only nondegenerate critical points. Then the Morse poly nomial dominates the Poincare polynomial; there is a polynomial Q (t) with nonnegative coefficients and �)n(t) - P (t)
=
(1
+ t) Q (t)
In particular we have the "weak " Morse inequalities
and equality Il
In particular,
Il
the total number of critical points on M is bounded below by the sum of all the Betti numbers.
386
HARMONIC FORMS
I n our toral example b a = 1 = b 2 and b 1 = 2, while ma = 1 , m 1 and �')ll t ) - p ( t
(
)
= (1
By writing out Q ( t ) = successively derive the
+ + 2t 2 ) 3t
'L::6 ql-. t l-.
- (1
with ql-.
+ 2t +
2 t )
+ t2
= t
=
and m 2 :::: 2
=
3,
(1
+ t)t
:::: 0 it is not hard to see that
We
can
Strong Morse inequalities ( 1 4.4 1 ) ma > ba
mn - mll - l
+
.
. . ± ma = bn - bn - 1
+
.
.
.
± ba
( 1 4 . 4 0 ) : For simplicity we assume that there is only one critical point at each critical level (this is generically so). At any level f = a (critical or not) we shall consider the space Ma , the Morse polynomial 91l(Ma ; t) , for this space, and the Poincare polynomial, again just for this space, and we shall observe how these polynomials change, /}. P, and so forth, as we pass through a critical point. It is clear, since topology changes only when passing through a critical point, that /}.�')1l and /}. P are nonzero only when passing through a critical point. Let �')ll ( t ) and P (t) have the value 0 on the empty set, that is, below the absolute minimum At the absolute minimum we have a point and its index is O. Thus on passing from the empty set to the set consisting only of the minimum point we have /}. 011 = 1 and /}. P = 1 . (We shall keep our toral example in mind.) As we continue to higher values of f we see the following. Consider passing though a critical point of index A at f = a , with its associated relative cycle, a disc el-. of dimension A . There are two possibilities:
PROOF SKETCH OF
1.
The boundary of this disc is a (A - I ) -cycle (sphere) that bounds in M;; . (In the toral example the boundary of the I -cell e 1 from the saddle at f = I is a pair of points that clearly bounds a I -chain on MI ' ) Let then a el-. = a cl-. where c lies on M;; . Then eA c). is an absolute cycle on Ma . It cannot bound in Ma ; if it did, el-. - Cl-. = a CH I would yield that el-. = a CH I + C and so el-. would be a trivi al relative cycle, a contradiction. A Thus in this case we have t.�)ll = [I-. and also t. P = (A and so t. (�')ll - p) = O . 2. The boundary of the disc is a (A - I ) -cycle (sphere) S on M;; that does not bound in M;; . But this says that S is a nontrivial (A - I )-cycle on M;; that bounds in Ma . Thus in this case t. 01l = [A and t. P = _ [ 1-. - 1 , and so t. (�')ll - P) = (A + tl-.- 1 = (l + t ) t H . These two cases show that on crossing a critical point of index A , �')ll - P change s either by 0 or by ( 1 + [ ) t l-. - 1 Since �)ll and P start out equal on the empty set we h ave demonstrated ( 1 4.40) . D -
•
BOUNDARY VALUES, RELATIVE HOMOLOGY, AND MORSE THEORY
387
Note th at in case ( 1 ) we can say that the relative cycle eA on M" mod M; is com pletable to an absolute cycle on Ma . In this case we have shown that � (,'Jll P) = o. -
thus
Corollary (14.41 ) : If all the relative cycles from all the critical points are com
pletable, then the Morse inequalities are equalities, m A = b A • In our toral example the 2-cell at level
f = 4 is the only relative cycle that is not This is reflected in m = 2 > b2 = 1 and m 1 = 3 > b 1 = 2. comple table. 2 If some critical points are degenerate, the Morse inequalities need not hold. In o Pr blem 1 4.3(4) you will study a smooth function on the 2-torus T 2 that has only 3 critical points. (Of course there are always a max and a min on any compact space.) A final comment. For a continuous, nondifferentiable function on a closed manifold M we still have the notions of the absolute maximum and minimum values, but we c annot talk about minimaxes since we don' t have partial derivatives at our disposal. Note, however, that we may define a homotopically critical value as earlier. We can also define a homotopic ally critical point to be a point y, at level fey) = a, such that some homology group Hj (M; U { y } , M; ) i= O. Problems
1 4.3(1 )
------
Consider a conducting su rface of genus 9 bounding a reg ion
M3
Figure 1 4. 1 2 Let there b e constant cu rrent loops in the exterior region , some of which th read through the holes. Assume that the appropriate boundary condition is that the normal component of B must vanish on the surface. Show that there is a unique static magnetic field inside M3 , determined completely by the currents in the loops that thread the holes.
1 4. 3(2) Show that
d sends normal forms i nto normal forms and that forms to tangent forms.
d* sends tangent
388
HARMONIC FORMS
/\
/\
be the normal forms, and let tan be the tangent forms. It can be nor shown that the global orthogonal decomposition that replaces the HOdge de com position ( 1 4.30) is
1 4.3(3) Let
P (P-1\1 ) (P+1\1 )
1\
=
d
+d
*
nor
+ harmonic p-fields
tan
Show that these subspaces are i ndeed orthogonal .
1 4.3(4) We have d rawn i n Fig u re 1 4. 1 3 a few level cu rves of a smooth functio n f on the torus J2 having a max at f = 2, a min at f -2, and a single other criti ca l point (at the fou r identified corners) at f = O. It is clear that the corner poi nt is =
critical since the level curves comprising must vanish there) .
f = 0 intersect there
(and so grad
f
j= O
j= O
Figure 1 4. 1 3 Continue this pictu re periodical ly in the plane s o that the corner point i s at the center. Show that from this center there are three directions for which the function decreases as one leaves the critical point, each pair being separated by a d i rection i n which the function is increasing. This shows that the critical point is degenerate. This critical point is of the type of a mon key saddle; see [M , p. 8] . Find two independent relative 1 -cycles in ( T02 ; Tl"- ) emanating from this critcal point. ( I n a sense, then , this critical point counts as 2 critical points of index 1 eac h . )
m)..+1
H1
1 4.3(5) P rove Morse's lacu nary principle: i f in t h e nondegenerate case w e have m).. - 1 =
0
=
7)1l(t)
-
pe t)
for some A, then m).. = b).. . ( H i nt: Write out the polynom ial eq u atio n (1 + t ) a( t) explicitly. )
=
P A R T THREE
Lie Groups, Bundles, and Chern Forms
CHAPTER
15
Lie Groups
15. 1 . Lie Groups, Invariant Vector Fields, and Forms
Is the unitary group S U (n ) connected?
lS.la.
Lie Groups
Let M (n x n) be the set of all n x n real matrices. As in Section 1 . 1 d, we shall associate to the matrix x the point in n 2 -dimensional euclidean space whose coordinates are XI I , X l 2 , . , xnn · Topologically then, M (n x n) is simply euclidean n 2 space ! The general linear group GI(n , JR) is the group of all real n x n matrices x = (xij ) with determinant det x i=- O. Since det x is an n th-degree polynomial in the coordinates, it is a smooth function on M (n x n ) . Since the real numbers differing from 0 form an open set in JR, and since the inverse image of an open set under a continuous map is open, GI(n , R) is an open subset of M (n x n ) . (This says that if det x i=- 0 then det Y i=- 0 if y is sufficiently near x .) Topologically GI(n , JR) is an open subset of euclidean space, and as such is an n 2 -dimensional manifold. It is clear from (x Y ) ij = L X i k Ykj that the product matrix has coordinates that are smooth functions of the coordinates of x and y . From the formula for the inverse .
.
X - 1 = --
X det x where Xu is the signed cofactor of x ji , and the fact that det x i=- 0, we see that the coordinates of x - 1 are also smooth functions of those of x . This leads us to the concept of a Lie group.
A Lie group is a differentiable manifold G endowed with a "product," that is, a map (g , h ) � g h
GxG� G
making G into a group. We demand that this map, a s well a s the "inversion map"
be differentiable.
391
392
LIE GROUPS
I n th e following examples, the reader should verify that th e g ive n mani fold s are i nd
groups. For example, GI(n, JR.) is a group because, first, det x =j:. 0 and det y :f:. 0 det (xy) = (det x ) (det y) =j:. 0, and second, det X - I = (det X ) - I =j:. O.
imp� es
Examples:
JR. , the additive group of real numbers . The product here is addition of real numbers . This group is commutative, or "abelian." JR. + , the multiplicative group of positive real numbers. This is ag ain abelian. G Gl (n, JR.) , the general linear group of all n by n real matrice s g w i th det G g =j:. O. Simil �rly, we have the nonsingul ar com ex mat� ces Gl(n , C). By writin g . . a 2n -dimensIOnal X j k t lYjb we see that Gl(n ,
1. G 2. 3.
=
=
t
=
=
4.
=
=
5.
R2 (e)
=
[
c �s e sm e
- sin e cos e
]
( 1 5.1)
and as such, 5 0 (2) is a curve parameterized by e in 1R4 , defined by X I I = cos 9, X l 2 = - sin e, X 2 1 sin e, and X22 = cos e. As a manifold this curve is diffeo morphic to the circle 5 I in the plane defined by X l = cos e and X2 = sin e , and this is the way we usually think of 5 0 (2) ; to a rotation of the plane through an angle e we associate the point on the unit circle 5 I at angle e . Sometimes we think of 5 0 (2) as the points exp(i e ) = eili of the complex plane. To compose two rotations eili and eirj) we simply multiply eiliei¢ = ei (liHl , that is, we add their angles . 5 0 (2) is abelian, whereas 5 0 (n ) , for n > 2, is not. 6. G = V (n ) , the unitary group, consisting of complex n x n matrices Z = (Zjk) with z t : = ZT = Z - I . The overbar denotes complex conjugation; the dagger denotes (hermitian) adjoint. The same type of argument that was used for O (n) in S ection 2 1 will show that V (n ) is a submanifold of complex n2 space or real 211 spac e, and is thus a Lie group. We easily see that det z has absolute value 1 . Note that V ( 1 ) is the group of complex numbers z = eie of absolute value 1 , and thus V ( l ) is isomorphic with 5 1 , that is, 5 0 (2) . V ( l ) is the only abelian unitary group. 7. G 5 V (n ) is the special unitary group; det z = + 1 . 8. G = T n i s the abelian group of diagonal matrices of the form =
=
z =
diag[exp (iel ) , . . . , exp (i ell ) ]
( 1 5 .2)
393
LIE GROUPS, INVARIANT VECTOR FIELDS, AND FORMS
This group is topologically S l x . . . X S l , the topological product of n copies of the circle, and as such is an n-torus. Since the circle is connected (each point can be l joi ned to the identity by a curve), it follows easily that Ti is connected. From this we may see that the far more complicated group U ( n ) is also connected ! Before doing so, we note the following. As a manifold, a Lie group is very special for the fol lowing reason . A Lie group wa al ys has two families of diffeomorphisms, the left and right translations. For g E G , these translations are defined by
Lg : G -+ G
an d
( 1 5.3)
R g : G -+ G
It is clear that the mapping inverse to L g is simply L g _ l . Theorem (15.4) : U (n ) is connected.
Note that Til is clearly a subgroup (and consequently a subset) of U (n ) . The familiar "principal axes theorem" of linear algebra states that any g E U (n ) can be diagonalized by a unitary matrix . (Proof: Each such g has eigenvalues of absolute value l . Let e l be an eigenvector with eigenvalue exp (i tl ) ) . Let et be the orthogonal subspace to e l in the Hermitian metric (v, w) = � vkuh . Since g is an isometry, g sends e t into itself and so g has an eigenvector e2 in this subspace with eigenvalue exp(itl2 ) . Continue with thi s process. In the eigenvector basis e l , e2 , . . . , en , the linear transformation g has matrix Z = diag(exp(itl) ) , . . . , exp (i tlll ) ) , as desired.) This means that given g E U (n) , there exists an h E U (n ) such that h - I gh = z = diag (ex p (i tl l . . . , exp (itln ) . Then g = hzh - I • (This says that g E (h T" h - I ) , i.e., g lies on the diffeomorphic copy of Til that results from left translating Til by h and then right translating by h - I .) Thus g can be joined to the identity by a curve, by putting tlj (t) = ( I t)ej . U (n ) is connected. D PROO F :
,
-
The subgroup yn of U (n) given by ( 1 5 . 2) is called a maximal torus of U (n) . Any conjugate h Tn h - I of this maximal torus is also called a maximal torus. By the same type of reasoning we may deal with the rotation group. Theorem (15.5) : 0 (n) consists of two connected " components " and S O (n) is
the component holding the identity. P R O O F : Consider first the case S O (2n ) . The principal axes theorem states that any g E S O (2n ) is "conj ugate" to a block "diagonal" matri x with 2 x 2 rotation matrices down the di agonal
( 1 5 .6) where R 2 (tlk ) is as in ( 1 5 . 1 ) . This simply says that after a suitable orthogonal change of basis in ]R21l the rotation takes on the form of rotations in n orthogonal 2-dimensional planes . In the case of S O (2n + I ) one adds a final diagonal ,
394
LIE GROUPS
entry o f + I . We can arrive at this canonical form a s follows. The possible real eigenvalues of g E S O (2n) are ± I , whereas the complex eigenvalues appear in complex conjugate pairs . If + 1 is an eigenvalue then it must be a double eigenvalue since det g = 1 . The eigenspace for this double eigenvalue is a 2plane E I on which g takes the form R 2 (0) . Likewise, if - 1 is an eigenvalue, it also must be a double root and we get a 2-plane E- l on which g takes the form R 2 (n) . In both cases g leaves i nvariant the orthogonal complementary ( 2n - 2) space. By continuing in this complementary subspace we either exhaust the entire 2n -space or have left a remaining 2k-space ]R 2k on which g has only complex eigenvalues. Let S 2k - 1 be the unit sphere in this subspace. The function f (x ) : = (gx , x ) takes on its minimum at some point Xo of the sphere. Now gxo does not lie along Xo since g has no real eigenvalue i n ]R 2k . We claim that the plane spanned by Xo and gxo is sent into itself by g . By definition, g sends Xo into this plane; where does it send gxo ? Let x (t ) be a curve on S 2k-l starting at Xo and put v = x ' (O) . Then 0 = f ' (O) = (gv, xo ) + (gxo , v) == ( v , (g T + g)xo) = ( v , (g- I + g)xo ) for all tangent v . Thus (g- l + g)xo = AXo and so g 2 xo = AgXo - Xo. Thus g sends g (xo) into the plane spanned by Xo and gxo , as desired. But it is immediate that g takes the form R 2 (8) on any invariant 2-plane. We may then continue with the complement of this plane i n ]R2k . Finally, in the case S O (2n + I ) , any g has + 1 as an eigenvalue, with a corresponding eigenvector. We proceed with the complementary ]R 211 as earlier. We continue with the proof of Theorem ( 1 5.5). The collection of all rotations of the form ( 1 5 .6) (with a+ I included in the odd-dimensional case) forms again an n -dimensional torus S I x . . . X S I , a maximal torus Til of the rotation group. One then proceeds as in the V (n) case to show that S O (n) is connected. o (n) consists of the rotations S O (n) and the improper orthogonal matrices O - (n ) where the determinant is l But if we let h = diag(- I , 1 , . . . , 1 ) E 0 - , then left translation L h by h is a diffeomorphism of O (n ) that interchanges S O (n ) and O - (n ) , showing that these two subsets are diffeomorphic. 0 -
.
Our final example, although not as intrinsically i mportant as the preceding ones, will play an important role in our treatment because it will be possible to perform explicit calculations. It is a nonabelian, noncompact, 2-dimensional Lie group. 9. G = A ( l ) , the affine group of the line, consists of those real 2 x 2 matrices
with x > O. The manifold for A ( l ) can be considered as the "right half plane," those (x , y) E ]R 2 , with x > O. A matrix group is a subgroup of GI(n ) that is also a submanifold of GI(n). All of our previous examples are groups of matrices. Although there are important Lie groups that cannot be realized as matrix groups, for our calculations we shall occasionally pretend that our group is indeed a matrix group, since the constructions and proofs are easier to visualize.
LIE G ROUPS, I N V A R I A N T VECTOR FIELDS , A N D FORMS
395
IS.lb. Invariant Vector Fields and Forms Lie groupS are special as manifolds for the following reason. Given a tangent vector Xe identity e, we may left or right translate Xe to each point of G , by means of to G at the the differe ntials ( 1 5.7)
resp. Xg
:=
Rg * Xe
y ielding two nonvanishing vector fields on all of G ! In fact, if we take a basis X" . . . , Xn for Ge (the tangent space to G at e), then we can left or right translate this basis to give n linearly independent vector fields, such as, ( 1 5 . 8) on all of G ! In particular,
every Lie group is an orientable manifold! Consider for instance, a closed orientable surface M2 of genus g. We shall see in Section 1 6.2 that of these surfaces only the torus (genus 1 ) can support even a single nonvanishing tangent vector field. In fact T 2 supports two vector fields 8/ 8e , 8/ 8¢, and the torus is indeed the commutative group S ' x S I with multiplication Topologically, the only compact Lie group of dimension 2 is the torus. (The Klein bottle is nonorientable and admits a nonvanishing vector field, but not two independent ones ! ) We shall say that a vector field X o n G is left (right) i nva ri an t i f i t is invariant under all left (right) translations, that is, resp .
( 1 5 .9)
You should convince yourself that if Xe is given, then ( 1 5 .7) exhibits the unique left (resp. right) invariant field generated by Xe . Similarly, for example, an exterior p-form a o n G i s left invariant if ( 1 5 . 1 0) and to get a left invariant form on all of G one translates a form at e over the entire group by (15. 1 1) In the case of a matrix group, Lg * Xh is especially simple. Let t H- h (t ) be a curve of matrices in G with h (O) = h and h ' (O) = Xh • Since G c GI (n) , this curve is simply a matrix h whose entries h j k (t) are smooth functions of the parameter t . h (t ) describes a curve in n 2 -dimensional euclidean space (real or complex). Then Xh , the tangent to this curve, is simply the matrix whose entries are the derivatives at t = 0, h j k (0) . There
396
LIE GROUPS
. f r�1I
is n o reason to believe that this new matrix h' associated t o the poi nt (m atri x ) h belong to the group G ( th i s will be illustrated in the case A ( I ) later). Then co n stan t matrix g , the curve t � g h (t) will have for tangent vector at t = 0 the L g . Xh =
o matri�
gh' (O) = gXh
that is simply the matrix product o f g and Xh . Example: G = A ( l ) , (Example (9». We may consider A ( l ) either as a sub man ifold
�4 or as the right half plane, since the entries 0 and 1 at the bottom contribute nothi n to our knowledge of the matrix. Since
of g
we see that the right half plane is endowed with a rather unusual multiplication given in the top row of this matrix equation. We shall identify
[� n
E
with (x , y )
A(1)
and for tangent vectors we identify
[�
.
dt
wIth
o
(
R2
E
)
dX
dy T
dt
dt
-
which is the tangent vector (dx /dt ) a/ax + (dy /dt ) a/ay . Now let us left translate the vectors a
and
ax
[
a ay
n
at the identity e to the point (x , y) . For a/ax we consider the curve h ( t ) given by 1
;
t
whose tangent at e is a/ax . Then, letting g be the matrix
we have a L g' ax
= - ( gh ( t ) } t = o d
dt
=[ ] x o
0
0
and this is indeed the left translate of a/ax at the identity to the point (x , y)
[� n [� � ]
(Note that this matrix is not in A ( I ) ; it is a tangent vector to A ( I ». Thus the left transl ate of a/ax to (x , y) is a X = x ax
397
LIE GROUPS, INVARIANT VECTOR FIELDS, AND FORMS
To co ns truct the
left translate of a / a y at (l , 0) to the point (x , y ) we form
The re su lt is the vector x a / ay. Thus a basis for the left invariant vector fields on A ( 1 ) is
given by the pair
a XI = x ax
( 1 5 . 1 2)
a X2 = x ay
Next note that in any Lie group, if X I , . . . , XI! is a basis for the left invariant vector fi eld s and if CJ I , . . . , CJ n is the dual basis of I -forms, then this dual basis is automatically
left invariant, since
L ; CJg ( Xe )
= CJg { L g * Xe } = CJg (Xg ) = CJe ( Xe )
that L ; CJg = CJe · The same argument shows that if a P is any p10rm whose values a is left invariant. on any p- tuple of left invariant vector fields are constant on G, then s Thu the basis of left invariant I -forms dual to ( 1 5 . 1 2) is given by shoW S
dx I CJ = x
CJ
2
=
( 1 5 . 1 3)
dy x
If a and fJ are invariant under left translations then so are d a and a !\ fJ . Thus in A ( 1 ) CJ
I
!\
dx !\ dy 2 CJ = ----'-
x2
is a left invariant area form or left Haar measure ; for any compact region U for any g E A ( l )
and
J1
gU
dx
� dy =
x
J1
U
dx
� dy
( 1 5 . 1 4)
C
A (l ) ,
x
where g U : = L g U is the left translate of the region U . This would not hold if the factor
x-2 were omitted.
Problems
1 5.1 (1 )
For the group A(1 ) , find the right i nvariant vector fields coinciding with 8/8x and 8/8 y at e , find the dual right invariant 1 -fo rms, and write down the right Haar measure.
1 5.1 (2) jR4 can be identified with the space of all real 2 x 2 matrices, identifying x = (x 1 , x2 , x 3 , x4) with the matrix (again called x)
81(2, jR) can be considered as the subman ifold M3 of jR4 defined by det(x) = 1 . 81(2, jR» acts linea r/y on jR4 , g : jR 4 -+ jR4 , by g(x) = gx (matrix multiplication) .
(i) Compute the 4 x 4 matrix diffe rential g* of 9 and show that det g* 1 . This shows that the action of G on jR4 prese rves the euclidean vol ume form . =
398
LIE GROUPS
det (X) i s of course a function on JR.4 that i s invaria nt under t action of G. Use Equation (4.56) to write down a left i nvariant vol um e for all of 81(2, JR.).
(ii) H(x)
3'fO�
:=
---
15.2. One-Parameter Subgroups Does eO} = (cos 9 ) / + (sin 9 ) 1 look familiar?
A homomorphism of groups is a function
J:G� H that preserves products In Section 1 3 . 1 , we defined the special case of a homomorphism when the groups were abelian, and when the group "multiplication" was "addition." As an example, the usual exponential function J (t) = e l defines (since eHI = eS er ) a homomorphism exp : lR. � lR. + of the additive group of the reals to the multiplicative group of positive real numbers. Note that exp is also a differentiable map, and in this case it is 1 : 1 ( the homomorphism is injective), and also onto (surjective). We then say that exp is an isomorphism of Lie groups. exp is a diffeomorphism with inverse log: lR. + � R A 1-parameter subgroup of G is by definition a differentiable homomorphism (in particular, a path)
g : lR. � G
t
�
g et)
E
G
of the additive group of the reals into the group G . Thus
g (s
+
t)
=
g (s)g(t )
=
g (t)g(s )
Consider now a I -parameter subgroup of a matrix group G.
Figure 1 5.1
( 1 5 . 15)
399
ONE·PARAMETER SUBGROUPS
+ s) = g (t ) g (s ) , that is, gij (t + s) both si de s with respect to s and put s = 0, A
s matrices g ( t
g ' (t )
S i nce g
= g et )
=
Lk gik (t)gkj (s) . Differentiate ( 1 5 . 1 6)
g ' (O)
'(O) is a constant matrix, the solution to this is g et )
w he re
= g (O) exp ( tg' (O) }
exp(5) = e S := I + 5 +
52
53
- + -
2!
3!
+
...
It can be sh own that this infinite series converges for all matrices 5. Since g (O) any homomorphism g : IR. -+ G, we conclude that g e t ) = exp { tg ' (O) }
( 1 5 . 17)
= e for ( 1 5 . 1 8)
is the most general form for a I -parameter subgroup of a matrix group G .
Equati on ( 1 5 . 1 6) tells us how to proceed even if G i s not a matrix group, for i t really
s ay s
g ' (t) =
Lg(t ) * g' (O)
( 1 5 . 1 9)
that is, the tangent vector X to the I -parameter subgroup is left translated along the subgroup. Thus, given a tangent vector Xe at e in G, the I -parameter subgroup o f G whose tangent a t e is X e is the integral curve through e of the vector field X on G resulting from left translation of X e over all of G .
The vector Xe i s called the
infinitesimal generator of the I -parameter subgroup.
F i g u re 1 5.2
e
For any Lie group G we shall denote the I -parameter subgroup whose generator at
is X, by
g et )
x : = el = exp tX
just as we do in the case of a matrix group.
400
LIE GROUPS
For example, i n A ( I ), to find the I -parameter subgroup having tangent vector (a b) T at the identity, we left translate this vector over A ( I ) .
( 1, 0)
Figure 1 5.3 The left translate of (a 8 I 8x + b8 I 8y) to the point (x , y) is, from ( 1 5 . 1 2), (ax 818x + bx 818y) . Then we need to solve
dx - = ax x (O) = I ( 15 .20) dt dy - = bx y (O) = 0 dt The solutions are clearly straight lines dy I dx = b I a , but to see the parameterizatio n
we must solve ( 1 5 . 20) to get
x (t)
=
ea t
y (t)
=
(which never reaches the y axis).
beal a
-
-
b
a
�
y
----r-----��-- x
Figure 1 5.4 In Problem
1 5.2(2) you are asked to get this from the power series.
401
O N E - P A R A M E T E R S U B G ROUPS
Problems
1 5 . 2( 1 ) We shall see in the next section that J=
[ � �1 ]
can be considered a tangent vector at the identity of the group G I (2 , = -I - J, J4 I to show
J2
,
J3
=
=
lR) . Use
,
[0 ] [
eO J
ex p
=
-Sin e]
(cos e ) 1 + (sin e ) J =
-e
e
O
cos e sin e
i. e . ,
cos e
for all real e . This 1 -parameter subgroup of G I (2, lR) is the entire subgroup of rotations of the plane, S O(2) ! Warning: It makes no more sense to say exp S + S for S small than it does to say eX = + x when x is a small number. For example, 1 + e J is never in S O(2) for any e # o .
1 5.2(2) Compute
1
=
I
directly from the power series.
1 5 .2(3) Consider the diffe rential equation x' ( t)
=
d
��t) = A( t) x ( t)
x (O) = Xo where A( t) is an n x n matrix function of t and x ( t) is a col u m n matrix . It is known that if A is actually a constant matrix, then the solution is x ( t) = exp( tA) xo ; this easily follows formal/y (i.e. , disregarding questions of differentiating infinite series term by term , etc. ) from the power series expansion of exp ( t A) . In the case of a 1 x 1 matrix function A( t) the sol ution is of cou rse x( t)
=
exP(
l
t
A(r ) dr ) xo
We claim that this same formula holds in the n x n case provided that the matrix A(t) commutes with its indefinite integral 8( t) : = J; A( r ) dr for all t. Verify this formally by looking at x ( t) : = ex p [ 8( t) ] xo = and using
B' ( t) = A( t) .
[I
+ 8( t) +
�
( 8 ( t) B( t) }
+...
]
Xo
402
LIE GROUPS
1 5.3. The Lie Algebra of a Lie Group
What is the third B etti number of the eight-dimensional SI(3, JR)?
15.3a. The Lie Algebra Let G be a Lie group. The tangent vector space G e at the identity e plays an important role; we shall denote it by the script �/
and call it (for reasons soon to be discussed) the Lie algebra of G . Let X R , R = 1 , . . . , N, b e a basis for � ; X R will also denote the left translation of this field to all of G . Since any left invariant vector field is determined by its value at e, the most general left invariant vector field is then of the form
where the v R are constants. Let a R , R = 1 , . . , N be the dual basis of left invariant I -forms on G ; they are determined by their values on vectors from � . The most general left invariant r-form on G is of the form .
It is again determined by its values on r-tuples from �/ . It is constant when evaluated on left invariant vector fields and al are constants. Recall the notion of Lie derivative or Lie bracket of two vector fields on a manifold M ; see Equation (4.4) .
Theorem (15.2 1 ) : The Lie bracket [X, Y j of two left invariant vector fields is again left invariant. A vector field X is left invariant iff 0' (X) is constant on G whenever a is a left invariant I -form. If a is a left invariant I -form then
PRO O F :
dO' (X, Y ) =
-0'
( 1 5.22)
( [X , V])
Since da is left invariant, the left-hand side is constant.
0
We may then write ( 1 5.23) for some structure constants Cks (dependent on the basis {X R } ) . I n Problem 15.3( 1 ) you are asked to prove the following.
403
THE LIE ALGEBRA OF A LIE GROUP
Theorem (15.24): The Maurer-Cartan equations
du v
=
=
L C�s u R !\ u S
-
R <S I 2
- -
L C�S U R R.S
!\
US
ho ld, and d 2 u v = 0 yields the Jacobi identity
C�S CfM
+ C�M C:L
+
C�L C:' s
=
0
This Jac obi identity for left invariant I -forms is also a consequence of a general Jacobi id enti ty for vector fields on any manifold Mil . If X, Y, and Z are any three vector fields o n a manifold, then as differential operators on functions f, [X, Yj (f) = X (Y (f)) Y(X( f )), and so on. Then the following Jacobi identity is immediate. -
[ [X , Y j , Z j
+
[ [ Z, X] , Yj
+ [ [V, Z] , Xj = 0
( 1 5 .25)
and in the case of a Lie group this gives ( 1 5 .24) via ( 1 5 .22). We now make the vector space �/ = G e into a "Lie algebra" by defining a product
�" x :r -+ �' as follows. Let X E �/, Y E � . Extend them to be left invariant vector fields XI, yl on all of G, and then define the product of X and Y to be the Lie bracket
[X, Yj : = [XI , y/]e This product satisfies the relation [X, Y] = - [ V, X] and the Jacobi identity ( 1 5 .25) . We shall see later o n that there are three vectors X, Y, Z in the Lie algebra of S O (3) that satisfy [ X, Y j = Z and [X, Z] = Y. Then [X, [ X, V]] = Y, while [[X, X] , Y] = 0, and thus the Lie algebra product is not associative ! We shall consistently i dentify the Lie algebra :r with the N ( = dim G) dimensional vector space of left invariant fields on G . Classically the Lie algebra �/ was known as the "infinitesimal group" of G , for classically a vector was thought of roughly as going from a point to an "infinitesimally nearby" point. �/ then consisted of group elements infinitesimally near the identity ! We shall not use this picture. -
-
15.3b. The Exponential Map Theorem (15.26) : For any matrix A, det eA = el r A . Consider the matrix A as a linear transformation of complex n-space = A v, then from the power series for eA we see that e A v = e A v . Thus eA has eigenvalues exp(A d , . . . , exp(A Il ) , where A I , . . . , A are the eigenvalues of A . Then, since the determinant is the product n of the eigenvalues PROO F :
en . If A is an eigenvalue of A, A v
det exp A =
II exp A; = exp L Ai
=
exp t r A
D
404
LIE G ROUPS
Theorem (15.27): The map exp : W --+ G sending A f---+ e A is a diffeomorphism of some neighborhood of 0 E �f0nto a neighborhood of e E G. We shall give two proofs. For a matrix group, look at the d ifferenti al of the exponential map applied to a vector X E 3"
PROOF :
d dt
d dt
1? 2 )
(
exp. (X) = - (exp tX)I =o = - 1 + tX + - t-X + . . .
2
=
X
1 =0 Thus exp. : 3' --+ �/ i s the identity and exp is a local diffeomorphis m by the inverse function theorem. If G is not a matrix group we would proceed as follows. Given X at e, e lx ::: exp(tX) is a curve through e whose tangent vector at t = 0 is the vector X ( recall that e X is the integral curve through e of the left invariant vector field X) . Thus again exp. (X) = X, and we proceed as previously. D
Remark: In a general Lie group, the I -parameter subgroup exp(t X) is the integral curve of a vector field on G, and thus it would seem that this need only be defi ned for t small. In this case of a left invariant vector field on a group, it can be shown that the curve exists for all t , j ust as it does in the matrix case. 15.3c. Examples of Lie Algebras
(n, JR.) . Let M (n x n) be the vector space of all real n x n matrices ; M (n dimensional Euclidean space. For A M (n x n)
1. G = 01
E
det e A = e trA
and therefore
Since dim
M(n n) = n2 x
>
x
n)
�
n2
0
M(n x 11 ) --+ 01 (n, JR.) dim GI (n, JR. ), we see that the Lie algebra o f OI (n , lR) �(n, JR.) = M(n x 11 )
exp : =
is
We shall now use the fact that if G is a matrix group, that is, a subgroup of Ol(n), then its Lie algebra , being the tangent space to the submanifold G of Ol(n , JR.) , is the W largest subspace of x such that exp : W --+ G . 2. G = S O First w e need two elementary facts about the exponential of a matrix. Since e A e - A = (I + A + A 2 / 2 ! + . . . ) ( 1 - A + A 2 / 2 ! ) = I we conclude
(n).
M (n n)
(e A ) - 1 = e - A
...
Next, from the power series it is evident that for transposes,
It is clear then that if A is skew symmetric, A T (exp A) - l
=
=
- A , then
(exp A ) T
405
THE LIE ALGEBRA OF A LIE GROUP
,A
and s o exp A E O (n ) . Also, since det e = e r = 1 for a skew A , we see e E S O (n ) . Thu s the skew symmetric matrices exponentiate t o S O ( n ) and the Lie algebra o f S O (n)
A
)li-C
A
n ) that contains the subspace of skew symmetric i s a vector subspace 6A}�n ) of matri ce s. A Co nversely, suppose that for some matrix A E �A1-(n) , that e E S O (n ) . Thus exp ( A )
Since exp is a local diffeomorphism it is e A is close enough to the identity then
= 1
exp ( - A T ) I
in a neighborhood of 0
E
�n ) . Thus if
that is, A is skew symmetric. Thus �.a(n) ,
the Lie algebra of S 0 (n ), is precisely the vector space of skew symmetric matrices.
n x
n
One can also see this by looking at the tangent vector to a curve g (t) in S O (/1 ) that ' ' 0, showing that g ' (O) is skew starts at e. Since gg T = e, we have g (O) + g (O l symmetric. 3. G = U (n ) , the group of unitary matrices, u - 1 u t , where t is the hermitian adjoint, that h -A, is, t e transpose complex conjugate . Then note that if A is skew hermitian, At then e A E U (n ) from the same reasoning. We conclude that =
=
=
u.(n) is the vector space of skew hermitian matrices. 4.
G
S U (n ) , the special unitary group of unitary matrices with det u = 1 . Since a skew hermitian matrix A has purely imaginary diagonal terms we conclude that det eA = e t rA has absolute value 1 . However if A also has trace 0 we see that e A will lie in S U (n) . =
�.u(n) is the space of skew hermitian matrices with trace 0
5. G
=
Given g
Sl (n , JR) ,
the real matrices g with det g =
1
a.Cc n , JR) is the space of all real matrices with trace 0
E
1S.3d. Do the 1-Parameter Subgroups Cover G?
G, is there always an
A E � such tha e A = g ? In other words,
is the map exp
t
:
W
�
G onto?
be shown that this is indeed the case when G is connected and compact. (It is that a I -parameter subgroup must lie in the connected piece of G that contains the identity.) SI(2, JR) is not compact. For g E Sl (2, JR) It can clear
g that i s locus
,
=
[: �]
xw
- yz
=
1
the coordinates x , y , z, w satisfy the preceding simple quadratic equation. This is not compact since, for example, x can take on arbitrarily large values. You are
406
LIE GROUPS
asked, in Problem 1 5 .3(2), to show that any g in S I (2 , JR) with trace < -2 is nev er of the form e A for any A with trace 0, that is, for any A E �£(2, JR) . This result is somewhat surprising since we shall now show that SI ( 2 , R) is con. nected !
g=
[ � �]
in S I (2 , R) can be pictured as a pair of column vectors (x z) T and (y w l in ]R2 sPannin g a parallelogram of area 1 . Deform the lengths of both so that the first b ecomes a unit vector, keeping the area 1 . This deforms S I (2 , JR) into itself. Next, "Gram-S ch mi dt" the second so that the columns are orthonormal. This can be done continu ousl y; instead of forming v - (v, e)e one can form v - t (v, e)e. The resulting matrix is th en in the subgroup S 0 (2) of S I (2 , JR); that is, it represents a rotation of the plane. We have shown that we may continuously deform the 3 -dimensional group SI(2, R) into the i -dimensional subgroup of rotations of the plane, all the while keeping the submanifold S O (2) pointwise fixed!
This last group, described by an angle e , is topologically a circle S ' , which is connected. This shows that S I (2 R) is connected. 0 In fact we have proved much more. Suppose that V k is a sub manifold of M n . (In the preceding SO(2) = V ' C M 3 = Sl (2, JR) . ) Suppose further that V is a deformation retract of M; that is, there is a continuous I -parameter family of maps rt : M -7 M having the properties that ,
1. ro is the identity, 2. r, maps all of M into V
3.
and each rt is the identity on
V.
Then, considering homology with any coefficient group, we have the homomorphism 1' , * : Hp ( M ; G) ----* Hp ( V ; G) , since r will send cycles into cycles, and so on; see ( 1 3. 1 7). If z p is a cycle on M and if r , (z p ) bounds in V , then z p bounds in M since under the deformation, z p is homologous to rt ( z p ) ; see the deformation lemma ( 1 3.21). Thus 1' , * is 1 : I . Furthermore, any cycle z � of V is in the image of 1' , * since z � = r, (z�). Thus r, * is also onto, and hence Theorem (15.28): If V
C
M
isomorphic homology groups
is a deformation retract, then
V
and
M have
Since S O (2) is topologically a circle S ' , we have Corollary (15.29) : Ho ( SI (2,
mology groups vanish.
JR) , :2)
�
:2
�
HI (SI (2,
JR) , :2) and all other ho
407
SUBGROUPS AND SUBALGEBRAS
Problems
--
15.3(1 ) Prove (1 5 . 24) . 1 5 3(2) Let A be real, •
2
mat rix says that
x
------
2 , with trace O. The Cayley-Hamilton theorem for a 2
A satisfies its own characte ristic equation
A2 - (trA) A + (det A) I
=
hen ce p :=
det
x
2
0
A
2 2
(The proof of the Cayley-Hamilton theorem for a x matrix can be done by direct calculation. One can also verify it in the case of a diagonal matrix, which is tri vial , and then invoke the fact that the matrices that can be diagonalized are "dense" in the set of all matrices, si nce matrices generically have distinct eigenvalues. ) Show that
(cos .fj5) I + (.fj5)-1 (sin .fj5) A
if
p
>
(cosh VIPI) / + ( VIPI)-1 (sinh VIPI) A
and, of course, e A
=
I
+ A if p
=
O.
g
is never of the form e A for A E 1 -parameter subgroup of SI(2,
15.3(3) (i)
=
if
Conclude then that
tre A
Thus, in particular
0
�
[ �2
p
<
0
-2
d
_O
6.f(2, JR.) . I n particular, this
g
does not lie on any
R).
Does SI (n, JR.) have an inte resting deformation retract? Is SI ( n, IR) con nected?
(ii) What are the i nteger homology groups of the 8-di mensional manifold SI (3, JR.) ?
(iii)
What can we say about G I ( n, JR.)? Is it connected?
1 5.4. Subgroups and Subalgebras
How can one find subgroups of G by looking at
�/?
15.4a. Left Invariant Fields Generate Right Translations
Let X
then
be a left invariant vector field on the Lie group G . If Xe is the value of X at e, exp (tXe)
408
LIE GROUPS
is the i -parameter subgroup generated b y Xe . We know that this curve is the integral curve of the field X that starts at the identity e. Since X is left invarian t, the i ntegral curve that starts at the generic point g E G must be the curve g et) : = Lg exp (t Xe ) :::: g exp ( t Xe ) On the other hand, X, as a vector field on a manifold G , generates a flow ¢t : G -+ G (at least if t is small enough), whose velocity field is again X. Thus it must be that ¢t (g) = g exp (t Xe ) Hence .
.
Theorem (15.30) : The flow generated by the left invariant field X is the 1-
parameter group of right translations ¢t (g)
=
g exp ( t Xe )
Figure 1 5.5 Since a right invariant vector field Y is then automatically invariant under the flow generated by a left invariant field X, we conclude that their bracket vanishes [ X1eft, y right ] = 0 ( 15. 1) 3
Of course, by the same reasoning, right invariant fields generate left translations.
15.4b. Commutators of Matrices
Recall that the Lie algebra ' as a vector space, is simply the tangent space to G at e, W but as an algebra it is identified with the left invariant vector fields on G . (Of course this is merely a convention; we could have used right invariant fields j ust as well.) If X E �, and Y E � � , then their Lie bracket
[X , Y]
=
·�xY E W
is given by the Lie derivative, or, as first-order differential operators [X , Y] ( f)
=
X (Y f) - Y(X f)
associated with the left invariant fields X and Y. If G is a matrix group, each X E is itself a matrix (not in G but rather in .the W tangent space to G at e). For example, we have seen that if G = S O (n ) then X IS a skew symmetric matrix. We claim then that [X, Y] is merely the commutator product
SUBGROUPS AND SUBALGEBRAS
of the ma trices
[X, Y]
=
X Y - YX
To see this we use Theorem (4. 1 2). We have, at e = I [X , Y]
=
lim
409
( 1 5.32) E
Gl (n , JR.)
{
t2 where
=
g exp(tX)
Thus
. {exp( tX) exp(tY) exp(- tX) exp(- tY) - l } [X, Y] = hm ( 1 5.33) 1 -> 0 t2 In Problem 1 5 A( 1 ) you are asked to show that this indeed does reduce to the commutator of the matrices. This shows, for example, that if X and Y are skew symmetric matrices then so is XY - YX.
15.4c. Right Invariant Fields All that we have said about left invariant fields can be redone for right invariant ones. Right invariant fields ("right fields" for short) generate left translations. We have defined the Lie algebra � to be essentially the vector space of left fields, and then [Xi , Xj ] =
xk Ct
What would this become if we had used right fields instead? Let {Xj (e) } be a basis for G e and extend them to left fields {Xj (g) } on all of G , X i (g)
=
L g * X i (e)
Let {Vi Ce) } coincide with the X's at e and extend them to right fields on G , Y i (g) = R � * V i Ce) = Rg * Xi (e)
X(g)
Figure 1 5.6
Y(g)
410
LIE GROUPS
We are interested in the "right" structure constants
We calculate these for a matrix group, though the result holds in general. The flow generated by Y i consists of left translations. Repeating the step s going into Problem 1 5 .4( 1 ) , but using right fields Y, we see
as matrices. We conclude (since Y = X at e)
and the right structure constants are merely the negatives of the left! By "Lie algebra" we shall always mean the algebra of left invariant fields.
15.4d. Subgroups and Subalgebras We are interested in subgroups of a Lie group. (We have already discussed I - parameter subgroups. ) For example SO(n) is a subgroup
SO(n)
c
Gl (n , JR)
of the general linear group and it is an embedded submanifold (we showed this in Section l . l d) . For a subgroup H c G to qualify as a Lie subgroup we shall demand that H , if not embedded, is at least an immersed submanifold. The 2-torus, consisting of points
is a 2-dimensional abelian group
with a I -parameter subgroup
where r and s are real numbers. As discussed in Section 6.2a, if s / r is irrational this curve winds densely on the torus; thus H in this case is an immersed, not embedded submanifold. This is not a closed subset of the torus since its closure (obtained by adjoining its accumulation points) would be the entire torus, but it still qual ifies as a Lie subgroup. The tangent space �ec n , JR) to Gl (n , JR) consists of all n x n matrices, whereas the tangent space .)Al(n) consists of skew symmetric n x 11 matrices.
41 1
SUBGROUPS AND SUBALGEBRAS
Figure 1 5.7 and Y be skew symmetric matrices. Left translate them over all of G I (n lP?) . Since the resulting vector fields are tangent at g E S O (n ) to S O (n) , so is their bracket [X, V] . In particular [X, Y]e E M�n) . This says that M�n) is not only a vector subspace of 3'f(n, lP?) , it is a subalgebra. Let X
,
In general, if H is a subgroup of G, then the Lie algebra � of H is a subalgebra of w. The converse of this is also true and of immense importance.
Theorem (15.34): Let G be a Lie group with Lie algebra �. Let � vector subspace 0/ � that is also a subalgebra Then there is a subgroup H
C
C
� be a
G whose Lie algebra is the given � C �.
For any n x n real matrices X and Y their commutator X Y - Y X has trace O. Thus the traceless n x n matrices form a subalgebra of �f{n, lP?) and there is a corresponding subgroup; it is, of course, Sl(n, JR).
Example:
Given the vector subspace � C �, left translate � over all of G, yielding a distribution D.. Let X l , . . . , Xr be left invariant fields spanning D. everywhere. Since � is a subalgebra
PRO O F :
Thus D. is in involution and is then completely integrable by the theorem of Frobenius. From Chevalley's theorem 6 . 6, we can construct the "maximal leaf" of this foliation passing through the identity ; that is, there is a manifold Vr and a 1 : 1 immersion F : v r ---+ G such that H := F ( V ) is always tangent to the distribution D. and passes through e E G. We claim that H is a subgroup of G ; that is, H is closed under the G operations a/multiplication and taking inverse.
412
LIE GROUPS
Let h I and h 2 b e in the leaf H . By the definition o f � , left translation of II by h I must send the leaf into another (perhaps distinct) leaf h I H of the fol iation, h I h2 E h I H .
Figure 1 5.8
However h I e = h I shows that h I is in both leaves H and h I H and since H is maximal it must be that H = h I H . In particular h I h 2 E H , as desired. A similar argument (Problem 1 5 .4(2)) shows H is closed under taking inverses. 0 Problems = X Y - Y X as matrices. (You needn't justify (legiti mate ! ) manipu lations with i nfinite series. )
1 5.4(1 ) Use ( 1 5 .33) and ( 1 5 . 1 7) to show [ X, Yj
1 5.4(2) Show that H is closed under taking inverses.
n
x n matrices ( At = A) with trace 0 form a subalgebra of iC) . Identify the subgroup. Is there a group whose Lie algebra consists exactly of the hermitian matrices?
1 5.4(3) Show that the skew herm itian
�n,
-
CHAPTER
16
Vector Bundles in Geometry and Physics
e
On the E arth's surface, the number of peaks minus the number of passes plus the number of pits is generically 2.
16. 1 . Vector Bundles What is a "twisted product"')
16.1a. Motivation by Two Examples
I. Vector fields on M. A section of the tangent bundle T M n to M" is simply a vector fiel d w on M. Locally, that is, in a coordinate patch ( U ; u l , , u" ) , w is given by •
•
•
its c omponent functions wt (u) , . . . , w� (u) with respect to the coordinate basis a/au, but of course these functions are defined only on U , not all of M . In another patch V , the same field is described by another n-tuple w � (v) , . . . , w� (v) . A t a point p in the overlap U n V these two n-tuples are related by where Cv u
3 v j 3 u is the Jacobian matrix. Thus a section of TM serves as a general ization of the ordinary notion of an n-tuple of functions F = (f l , . . . , f" ) : M N � JR." =
defined on an n-manifold, where now we assign a different n -tuple of functions in each patch, but we insist on a recipe telling us when two n -tuples are describing the same
"vector" at a point common to two patches. The bundle TM is, in a sense, the home in which all the sections live. Not all n-tuples are to be considered as tangent vectors, for there are other bundles "over" M. The cotangent bundle T* M uses a different recipe; its C v u is [3u j 3 v] T.
2 . The no rmal bundle
to the midcircle of the Mobius band.
Consider the Mobius band Mo 2 (a 2-manifold whose boundary is a single closed curve) and the midcircle submanifold M I = 5 1 . We are interested in the collection of all tangent vectors to Mo along 5 1 that are normal to 5 1 . We shall call this collection the normal bundle N ( 5 1 ) to 5 I in Mo. Clear! y we have a map that sends each normal vector to the point in 5 1 where it is based. 413
414
V E C T O R B U N D L E S IN G E O M E T R Y A N D P H Y S I C S
v
--------+-�--��-
U ----1
r-- U ---"
Ml = SI
Figure 1 6. 1 I t should b e clear that w e cannot fi n d a continuous normal vector field to S l that is everywhere nonzero, since if it points down at the left endpoint it must point up at the right endpoint because of our identifications. We have illustrated this with the nonnal field 'I! . If we wished to describe this field by a "component" I¥ we might proceed as follows. Select (arbitrarily) smooth nonvanishing normal vector fields e u and e v over patches U and V of S ' , U and V being chosen so that their union is all of M. Then at any point p E U n V we have
e v (p)
=
e u (p)c u v (p)
where C u v is a smooth nonvanishing 1 x I matrix defined in u n V . Note that this is the same notation that we used when talking about the tangent bundle; see equation (9.44). Also note that we may describe the nonvanishing of the "matrix" C u v as s ay i ng that
cu v
:
un
V
---*
GI O , lR)
Let 'I! be a smooth normal field to S ' . Then in U we have 'I! (p) = e u (p )1/!u ( p ) and in V, 'I! (p) = ev (p) 1/r v (p) , for smooth functions 1/r u and 1/r v . In the overlap
'I! (p) = e u (p) 1/r u (p)
=
e v (p) 1/r v (p)
and so
1/r v (p)
=
c vu (p) 1/ru (p)
where Cv u = cu� . Thus a normal vector field to S ' C Mti is described not by a single "component" function 1/r on S ' , but by a component function 1/r u in U and by a component function 1/r v in V, both related by the transition matrix C vu . , . Note that the local fields e u and e v allow us to say that the normal bundle N( S ) IS locally a product, in the following sense. The part of the bundle consisting of vectors based in the patch U is diffeomorphic to U x lR under the map
normal
415
VECTOR BUNDLES
N
(S I ) defi ned by u (p, l{f) = e u (p) l{f · Similarly I of N (S ) based in V into a product.
v : V
x
jR
-+
N ( S I ) makes the
�lthOugh N ( S lI ) is locally a product, it is globally twisted, for the entire N (S I ) is not .!Self product S There is no continuous way to assign a uniqueI normal vector ( p , l{f) for l{f a fixed real number, as p ranges over all of S . N (S is thus a
I to
a a pair
x
lR.
I)
I twiSted product of S and lR.
manifold If we were to consider the vectors normal to a curve M I in. , ae�_lRiemannian in each patch U of
", uld have to find (n 1 ) local normal fields ef , . . W we wo I, n field \]! would then be described by an (n M a d a normal -
1 )-tuple of components We shall consider this in Section 1 6. 1 d. I . � . fb' . . , l{f alize the notion of a K -tuple of functions on Mn we introduce the general To gener notion of a vector bundle over M . -
16.1b. Vector Bundles
A (re al or complex) rank K
vector bundle E over a base manifold Mn consists of a manifold E (the bundle space) and a differentiable map, projection such that
E is a local product space in the following sense. E
rr - 1 U
rr ------ ----- ----- M u
Figure 1 6.2
There is a covering of Mil by open sets { U, V, . . . } . There is a K -dimensional vector spa ce (the fiber) JR K or C K (and for definiteness we shall assume it to be JR K ) equipped with its standard basis "1 ,
.
..,
QK
We demand that for each open set U in the covering
U
x
jR K is diffeomorphic to the "part of the bundle over U ,
"
1T
-IU
416
VECTOR BUNDLES I N G EOMETRY AND PHYSICS
that is, there are diffeomorphisms
: V
x ]R K
�
JT - 1 V ( 1 6. 1 )
(In the case of the tangent bundle E = T Mil. , if eU is a frame in V then ( p. y) ::.:: i '" · represented , via . 'Vu , by a point i L- I � i � K e iU Y . ) A po m t s E JT - 1 ( V ) th en V and a K -tuple of real numbers y, the latter being the fiber coordinates of p� JT (s) E ( V n V ) , we demand that the two sets of fiber coordi nates be related by a nonsingular linear transformation
..... 1
IS
U
s. ' "VI"
c vu (p) : V n V
�
GI( K )
that depends differentiably on p =
Yv
that is,
=
i
Yv
c v u (p) y u
( 1 6.2)
c vu (p) ij Y uj
Note that each fiber over p, JT - I (p), is a K -dimensional vector space but it is not identified with ]R K until the patch V holding p is specified; only then can we use �-Iu to make the identification. (In the tangent bundle we can not read off the components of a vector until we have picked out a specific frame. ) Note also that in the name "rank K vector bundle," K refers to the dimension of thejiber, not the bundle space E. A (cross) section of E is a differentiable map s : M �
E
such that s (p) lies over p, that is, identity : M M Locally, over V , one describes a section by giving its vector components yu (p). subject to the requirement ( 1 6.2) in an overlap. In a triple overlap we have JT
0S
=
�
s
y w (p)
and so C wu and
=
=
c w v (p) Yv (p)
=
c wv (p)c vu (p)y u (p)
C w v C v u . Thus the transition functions {c vu } satisfy c vu (p) = c U V (p) - 1 c w v (p)c vu (p)c u w (p)
=
( 1 6.3)
I
Conversely, let M be a manifold with a covering { V, . . . } , and suppose that we are given matrix-valued functions C vu in each overlap c vu V n V Gl ( K ) that satisfy ( 1 6.3). Then we may construct a vector bundle over M whose transition functions are these C vu as follows. Take the disjoint collection of manifolds { V x ]R K , V ]RK , . . . } :
�
X
417
VECTOR BUNDLES
f r each o
patch. These are to be considered disjoint even though the patches can Now we make identificatio ns: (p , Yu ) E (U x ]R K ) is to be identified with (p ', Y v ) E ( V x ]R K ) iff p' = p and Yv = cvu ( p )Yu can be shown that the resulting identification space £ is indeed a K -dimensional or bundle over M with {cv u } as transition matrices. This is the procedure we used ction of the tangent bundle; from ( ax� /axt ) xt we see, from for constru t (16.2), tha
:lap. �
X�(x)
cvu (x )
=
=
ax v ax u
( 1 6.4)
Tangent bundle T 00 the other hand, for the cotangent bundle, a i' = a y ax{ /ax� = L j [(a x u /a v ) ] ij a y shows that
xT
cvu (x )
=
a )-I]T ( axax� )T [ ( a;: =
( 1 6.5)
Cotangent bundle T * M Two bundles whose transition matrices are inverse transposes are said to be dual vector
bundles. If E and £' are vector bundles over the same base manifold M , then the tensor product bundle £ 181 £' is defined to be the vector bundle with transition matrices Cvu ® c� u ' This means the following. A point in has vector components Yu = (yb , . . . , yt ) , a point in (U) has vector components Z u = (zh , . . . , z t ) , and a point in the tensor product bundle has the K L vector components
n- 1 (U)
n'- l
(Yu 181 Z u ) : = y� Z� i Ol
and
( 1 6.6)
by definition (cvu 181 c� u ) ( Yu
181
zu )
:=
(cvuYu ) 181 (c� u z u )
For exarnple, the mixed tensors, once contravariant and once covariant (i.e. , the linear transformations), form the vector bundle T M 181
T*
M.
16.1c. Local Trivializations
:
�
a product manifold. The diffeomorphisms ¢ u U x ]R K 11r- (U) that exhibitis locally the local product structure allow one immediately to exhibit K A bundle space £
: = ¢ u (p, eu )
over U, where again C-1 = ( 1 , 0, . . . , Ol , and so on, and these sections are linearly independent in the sense that at each p E U the vectors eOi ( p ) in the vector space (p) (the fiber over p) are independent. The eOi form a frame of sectio ns eOl ( p)
sections.
n-1
Note that one frequently proceeds in the reverse direction. For example, we made the collection of vectors normal to the midcircle of the Mobius band into a rank I vector
418
V E C T O R B U N D L E S IN G E O M E T R Y A N D P H Y S I C S
bundle by first picking out distinguished "sections"; this then defined the rna 41 general, suppose that we have two manifolds E and Mn and a map Jr E of III onto M. Suppose that each Jr - I (p) is a vector space ]R K . Supp ose that is a covering { U , V, . } of M and there are smooth maps e� : U E , :::: 1 ' such that Jr 0 e� is the identity map on U and the eu (p) are independent for eac h f( Define then
p� further � Th� en inC:; :
�
.
---+
.
-+
Q"
. . . •
---+
Yv
=
cvu (p)yu : =
0
u
( 1 6.2) is then automatically satisfied and we have made E into a vector b ndle over Mn and the e� yield a frame of sections over U . 0 We shall frequently denote a point of M by x , rather than p ; we are not implyin that x is a local coordinate, though that will often be the case. The most general
section over U is then of the fonn
where the
1/15 (x ) are component functions. We abbreviate this with lJI (x ) e U (x) 1/Iu (X )
=
em!
matrix notation
e U (x) 1/Iu (x )
=
( e f (x) , . . . , e � (x ))
[ 1 1/I (x )
: 1/ID� (x )
=
If lJI is a cross section over U n V , then in U n V we have lJI (x)
=
1/1
e U (x) 1/Iu (x)
=
e V (x ) v (x)
1/Iv (x )
=
cvu (x ) 1/Iu (x)
( 1 6.7)
If we can find a frame e of sections over all of M, we say that the bundle is a product
bundle, or is trivial. In this case
1/1 1 ,
.
.
•
,
1/I K )
k
=
L eu (x ) 1/I u u=1
yields a diffeomorphism
---+
E
making E globally a product manifold. In particular, a I -dimensional vector bundle (a line bundle) with a single nonvanishing global section is a trivial bundle.
VECTOR BUNDLES
In a nontrivial
419
bundle, the maps
:
=:
Jelizable .)
Note that every vector bundle E has in each U by 1jr 1 Cx ) = 0 , . . , 1jr K (X) die I l i ne proof. .
a global section, the zero section, defined locally = 0. In Problem 1 6. 1 ( 1 ) you are asked to give
-
16.1d. The Normal Bundle to a Sub manifold
Consider a Riemannian manifold V"+ K and a submanifold Mil C V. We define the DOrIDal bundle N(M) to M in V to consist of those tangent vectors to V that are based on M and are orthogonal to M.
Figure 1 6.3
(In the
figure, M is drawn as a curve.) It should be "clear" that if U c M is small one can find K smooth fields n Y , . . . , n � of normal vectors to M that are linearly independent at each point of U . Then, if
enough
denotes the
rr :
E = NCM" )
�
M
normal bundle
is again defined
by
K
420
VECTOR BUNDLES IN GEOMETRY AND PHYSICS
The A's are the components of a normal vector in the patch V . In a patch V would have a new frame {n� } and in an overlap V n V the frames would be relatedWe a K x K matrix function n v = nU C u v , and a normal vector would have two by components A u and A v related by A v = c v u A u , where c vu (x ) = c u Iv (x) E GI (KsetsItor) If we had chosen the frames nU and n V to be orthonormal, then c u v (x ) E O (K ) ' For example, the normal line bundle to the 2-sphere M 2 = S 2 C R3 ill trivial, N (S 2 ) = S 2 lR, since we have a global nonvanishing section given b Y outward-pointing unit normal. As we have seen, the normal bundle to the central circle M = S l of the MObi US band V 2 is not trivial. The normal bundle N (S I ) to the indicated circle SI C JR. p 2 is clearly itself an infinite .
u:
:::::
.
X
s'
Figure 1 6.4
Mobius band (the lengths of the vectors are not bounded). For this SI C �p 2 , N(S' ) is not trivial. If we use as model of JR.p 2 the disc with antipodal points identified, this S I can be deformed into a diameter. N (S is not trivial.
I)
Figure 1 6.5
Let C, x = x (t ) , 0 :s t :s 1 , be a closed curve in JR. 3 . Its normal bundle is a rank-2 vector bundle over C . Pick an orthonormal frame n = nCO) of two normal vectors na at p = x(O) . Transport this frame continuously around all of C, always remaining orthonormal and orthogonal to C , arriving at p x ( l ) but with perhaps a different frame n ( l ) from the original. Since JR.3 is orientable, and since the tangent T has retumed to itself, it must be that nCO) and n( l ) define the same orientation in the n ormal plane at p . This means that nCO) and n ( l ) are related by an S O (2) matrix g , nCO) 0 (1 )8· We are now going to redefine the normal framing along the last E seconds of the curve so that the framings match up at t = 0 and t = 1 . Since SO (2) is connected we can find a curve of 2 x 2 matrices g = g (s ) , 1 E :s s :s 1 , in S O (2), such thai =
=
,
-
P O I N C A R E' S T H E O R E M A N D T H E E U L E R C H A R A C T E R I S T I C
(1 - €) ::: I and g ( 1 ) = g. Now g tting ru es ) = n(s)g(s), yielding
421
redefine the normal frame on the last part of C by a framing with agreement at t = 0 and t = 1 . (By the curve g (s) to have s-derivative 0 at s = 1 E and at s = 1 we can even fra e e � th ming smooth.) The normal bundle to a closed curve in 1R3 is trivial!
�sing
-
P roblems
--
1 6. 1 (1 ) Show that the zero section is indeed always a section. 3 16.1 (2) IR p is the solid ball with boundary points identified antipodally. I s the normal bun dle to the circle S 1 c p 3 trivial?
Figure 1 6.6
1 6.1 (3) Is the normal bundle to
IR P2
in IR p 3 trivial?
Figure 1 6.7
1 6. 1 (4) Is the normal bundle to a closed curve in an Mn trivial? (Consider the cases M orientable and M not orientable. )
16.2. Poincare 's Theorem and the Euler Characteristic
Can you comb the hair on a sphere so that the directions vary smoothly and such that no hair
sticks straight out radially?
Before di scussing further properties of general vector bundles, we shall acquaint our selve s w ith the most important result on the sections of the tangent bundle to a surface.
422
VECTOR BUNDLES IN GEOMETRY AND PHYSICS
For further discussion the reader may consult Arnold's book on differenti al equations [A2, chap. 5]. 16.2a. Poincare's Theorem
Let M 2 be a closed (compact without boundary) surface and let v be a tangent vector field to M having at most a finite number of points p where the vector field Vanish yep) O. Generically this is so for the following reasons. The vani shi ng of a v field requires, locally, the simultaneous vanishing of two functions v and v 2 of the tw coordinate variables x and y , and generically these two zero sets intersect in iSOla points. Compactness (as in the proof of Theorem (8. 1 7)) then demands that there be only a finite number of zeros. Let p be a zero for v. We may assume that p is the origin of a local coordinate system x , y. Let S be a small coordinate circle, x 2 + / = E 2 , where by " small" we mean that p is the only zero inside S. Introduce a Riemannian metric in the coordinate patch. For example you may wish to use ds 2 = dx 2 + dy 2 . We may orien t the patch by demanding that x , y be a positively oriented system. We may then consider the angle that v makes with the first coordinate vector ax = a/ax at each point (x , y) on S ( an v ) e (x , y) = L. (ax , v) : = cosI I ax I I I I v I I We then have the following situation. Let So be a unit circle in an (abstract) ffi?; So is parameterized by an angle . We then have a map S -+ So defined by (x , y) the preceding angle e (x , y). This map has a Brouwer degree, called, as in S ection 8.3d, the (Kronecker) index of v at the zero p, written jv(p) = j (p). Of course it simply represents the number of times that v rotates as the base ofv moves around the circle S I . As such 1 j (p) _ de (x , y) ( 16.8) 2n Js In Section 8.3d we have illustrated the indices of four vector fields at the origin of =
1
1{
ee:; r.e:
}
=
=
M2
=
r
]R 2 .
We have made several rather arbitrary choices in the previous procedure, a Rieman nian metric, a coordinate system, and a closed curve S in the patch enclosing the zero. But the index varies continuously with the choices, and since it is an integer, it is in fact independent of the choices. In particular we may replace the circle S by a piecewise smooth triangle enclosing the zero. Note that we may compute the index even when the field v does not vanish inside the curve S, but the index will then be 0; see Problem 8.3(9). Finally, note that we may also consider a vector field that is smooth in a region except for an isolated "singular" point p; for example, the electric field grad(1/r) of a charge in ]R3 is smooth everywhere except at the charge. By the same procedure as at a zero, we may again define the index jv (p) of the vector field at the singularity. By a singularity of a vector field v we shall mean any point at which v is not s mooth or at which v = O.
P O I N C A R E' S T H E O R E M A N D T H E E U L E R C H A R A C T E R I S T I C
423
a s mooth vector field is not a singularity in the ordinary sense. In our present is called a singularity because the direction field defined by the vector is a zero. at ed undefin
�tuatiOn it zero o
f
poincare's Theorem (16.9): Let v be a vectorfield with perhaps afinite number o/ singu la ri ties on a closed surface M 2 . Then the sum of the indices of v at the singular points is in /act independent 0/ the vector field and is a topological invariant X .
discussed in the next section, X will be called the Euler characteristic. looking at the proof, let us look at some examples on the 2-sphere. The vector field a / ae tangent to the lines of longitude on the 2-sphere has a singularity at the north and poles. At the north pole the field looks like the "source" in Section 8.3d of w the south pole is a "sink," also of index 1 . Thus X (S 2 ) = 2 in this case. We hile 1 index the vector field a / a¢ tangent to the parallels of latitude, again with consider can also si gulari i e s at the poles. The indices are easily seen again to be both + 1 , verifying the theorem. Poincare's theorem implies the following, which we have mentioned many times the past: For reas ons Before
south
n t in
Corollary (16.10): Every vectorfield on S 2 has a singularity. Thus every smooth section of the tangent bundle of the 2-sphere must be zero somewhere, and hence this bundle is not a product bundle.
Thi s h as been paraphrased as "You can't comb the hair on a 2-sphere."
In our two fields a / ae and a/a¢ on S 2 , both fields had two singularities. We shall ex b t a field on S 2 with a single singularity (zero) with, of course, index +2.
now hi i
Figure 1 6.8
is obtained from a parallel field a/au in the u, v plane by stereographically projeThisctingfield(from the north pole) the field onto the tangent sphere. We have drawn the
424
VECTOR BUNDLES IN GEOMETRY AND PHYSICS
integral curves rather than the vector field itself. At the right of the fi gure W ha shown a view from the top, and one easily sees that the index at the north pole is +2. We can investigate this analytically as follows. Consider the sphere as the Riemann sphere, as in Section 1 .2d. In the com I plane C tangent to. the sphere at the south pole, we have the velOCI. ty fi eld of thpeeftx�w dw /dt = 1 , that IS, du/dt = 1 and dv /dt = 0. When we stereographically project tbj flow onto the Riemann sphere we get the parallel-like flow near the south pole w OS. Near the north pole z = ° we get
� ==
�;
(:�) (��)
=
-
As we go around the path z = e i e about z circuits, yielding the desired index 2.
=
=
( �2 ) 0,
=
-Z 2
the vector - Z 2
=
- eM
makes 2
The following proof is due to Heinz Hopf, who also proved the higher-dimensional version. We shall discuss this in Section 1 6.2c. We shall first prove the theorem in the case when M 2 is orientable; in the following section we shall then discuss briefly the nonorientable case. Choose any Riemannian metric for all of M 2 (see Section 3.2d). Let v and w be two vector fields on M, each having a finite number of singu larities. Some singularities of v may coincide with those of w. We know that M can be triangulated (see Section 1 3.2c). By choosing the triangles to be very small (e.g., by subdividing them) and by moving them around slightly, we may insist that (i) each triangle lies completely in some coordinate patch (xex , y",) ; (ii) the singularities of v and w lie in the interiors of triangles, not on edges or vertices; and (iii) there is at most one singularity of v and at most one singularity of w in the interior of any triangle. Then if I:!. is a triangle lying in a patch (xex , y", ) , we have the Kronecker index integers PROOF OF POINC A RE ' S THEOREM:
and 1 _ 1 dew 2n !a ,., where ev (x", , Yex ) = L ( a / ax", , v) and ew (xex , Yex ) are computed with the chosen Riemannian metric. Note that if I:!. lies in two patches, both coordinate systems will yield, as we know, the same indices. Then jw ( l:!.)
and
:=
P O I N C A R E' S T H E O R E M A N D T H E E U L E R C H A R A C T E R I S T I C
425
indices for the two vector fields, since, for example, if v has sumsrityofinthe then Thus their difference is :0 singula X X L j {d y _ 21][ 6 h 6 e
the
�
jv ( � )
v -
=
O.
e - dew }
w =
Now f:Jv(x" , yO' ) and ew (xO' , yO' ) depend strongly on the coordinate patch used. v
Figure 1 6.9
For example, if �O' and � fJ are adjacent triangles in patches (x" , yO' ) and Yp), then the angle that v makes with the first coordinate vector a / axO' is different from the angle it makes with the first coordinate vector a / aXfJ . However, (xfJ ,
ev (xO' , yO' ) - ew (xO' , yO' ) =
L (w, v)
f3
is the same as the difference constructed in the patch, since the preceding difference is merely the angle from w to v, which is determined by the Riemannian metric, independent of patch! Taking the differential of both sides
is a well-defined
Then
dey - dew
=
dL(w, v)
I -form on each edge of each a � , independent afthe patch used.
1 L j dL(w, v) Xw = _ 2 ][ 6 ];1 6 Since M is assumed orientable, we may assume that the coordinate patches have positive overlap Jacobians, and thus adjacent triangles �O' and � fJ will have the same orientation. But then L j6 dL(w, v) = 0 6 h becaus e each common edge will be traversed twice in opposite directions. Thus Xw , as desired, and their common value will be called X (M). Xv Note that if F M2 � V2 is a diffeomorphism, then F* will take the vector field v on M into a vector field F v on V, and it is easy to see that the index * of v at p is the same as the index of F v at F ( P ) . Hence X (M) = x C V) is a * diffeomorphi sm invariant. 0 Xv
-
=
:
We
shall now see how this integer is related to the topology of M 2 .
426
VECTOR BUNDLES I N GEOMETRY AND PHYSI CS
16.2b. The Stiefel Vector Field and Euler's Th eorem
We now know that we may evaluate X (M 2 ) on any closed orientable sUrface by I at any vector field with a finite number of singularities and summing the indices tiefel constructed the following vector field on any M 2 . Take again a triangulation of M 2 . Imagine that M 2 is the sea level surfac e of a pI . anet, we shall now construct a mountain range on the planet.
��ng .
.
Figure 1 6. 1 0
Put a mountain peak of height 2 at each vertex, a pit at the "midpoint" of each triangle at sea level 0, and a mountain pass of height 1 at the midpoint of each edge. The height of the land above sea level then defines a function on M2, and if are careful there will be a maximum at each vertex, a minimum at each face midpoint, and a minimax (saddle) at each edge midpoint. In the right-hand of the figure we have drawn the gradient lines for this function. The gradient vector has a zero at each peak, pass, and pit, and the indices there are + 1 , - 1 , and + 1 , respectively. Thus for this vector field X = no. peaks - no. passes + no. pits and we have proved
we
Euler's Theorem (16.11): For all triangulations of the closed M2 we have that the Euler characteristic
X (M 2 )
:=
no. vertices - no. edges + no. faces
is independent of the triangulation.
From the triangulation of the 2-torus in Section 1 3.3a we see that X (T2) O. Thus it would not contradict Poincare's theorem if there were a field on the torus with no singularities, and of course there is, v = 8/80 . We conclude with three brief remarks. Consider the projective plane lRp 2 . It is nonorientable, but it is "covered" twice by the orientable 2-sphere, since lR P 2 is S2 with antipodal points identifi ed. shaIl tbal lRP 2 discuss coverings more in Section 21 .2.) Thus we have a 2 : 1 map Jr : S2 locally is a diffeomorphism. ==
�
(We
P O I N C A R E' S T H E O R E M A N D T H E E U L E R C H A R A C T E R I S T I C
427
Figure 1 6. 1 1
Consider any vector field v on lR: p 2 . There is a unique vector field w on S 2 such that 1C. W v. In the figure, lR:p 2 is the upper hemisphere with antipodal identifications 2 on the equator, and v is the vector field on lR:p that rotates around the "north pole" 2 south pole on lR:p ). The field w rotates around both poles on S 2 . The ther ( e is no 2 singularity on lR:p at the pole has index + I and it is covered by two singularities on 1 and I: iw 52, each with the same index + 1 . Thus I: iv 2. On the other hand, 2 that if we take a triangulation of lR:p where each triangle is small, in ent it is evid each triangle will be covered by two disjoint triangles on S2 , then the that the sen se Euler characteristics, computed via vertices, edges, and faces, as in ( 1 6. 1 1 ), will satisfy 2 X (S 2) = 2X (lR:p 2 ) . Thus Poincare's theorem holds on the nonorientable lR:p 2 also 2 and X (lRP ) = 1 . This illustrates a general fact (discussed in Section 2 1 .2d): :::
=
=
=
Mil
Each nonorientable manifold has a "2-sheeted" orientable covering manifold whose Eu ler characteristic is 2 X ( Mil ) .
This allows us to prove Poincare's theorem for nonorientable surfaces as well. Second, Hopf has proved the n-dimensional version of Poincare's theorem. To a vector field v on an Mil with an isolated singularity p, we may again assign an index j (p) by taking a small (n - I )-sphere and considering again the Kronecker index of v on this sn- I . We may look at a triangulation of Mn and define the Euler characteristic X
(Mn)
=
( n o O-simplexes) - (no. I -simplexes) + (no. 2-simplexes) . . . + ( - I t (no. n-simplexes) .
-
and
we again have
Hopf's Theorem (16.12): For any closed Mn and any vectorfield v on Mn with isolated singularities, we have I: iv ( p ) = X ( Mn). The proof is considerably
more difficult (see [ G , P] or [M2]) Finally, a necessary condition for there to exist a vector field on Mn without any singu 0 larities is cle arly X (Mn ) = O. Hopf has also shown that this is sufficient; if X (Mn ) !hen there is some v on Mn with no singularities. One may again consult [M2] . =
428
VECTOR BUNDLES IN GEOMETRY AND PHYSICS
P roblems
Let
Mg
--
be a su rface of genus g. Let it stand on a ta ble an d let h be the
Figure 1 6. 1 2
function o n Mg measuring the height above the tabl e . By looking at th e vector field grad h on Mg , show that
X ( M�)
=
2
-
2g
1 6.2(2) Consider a function with only nondegenerate critical points (in the sense of
Morse, Section 1 4.3e) on a surface M2 . Its gradient vector at a critical point has Kronecker index 1 , 1 or 1 if it is, respectively, a minimum, saddle, or max. i m u m (see Figure 8.9). Show that the Poi ncare-Stiefel pits - passes + peaks theorem , together with Problem 1 6. 2 ( 1 ) , yields Morse's equality in Theorem 1 4.40. -
,
16.3. Connections in a Vector Bundle How can the tangent bundle to an orientable surface be considered
a
complex line bundle?
16.3a. Connection in a Vector Bundle
Let Jr E --+ be a rank-K vector bundle (real or complex). We shall introduce the concept of a connection for such a bundle by imitating the procedure used in Section 9.3 for the tangent bundle. A section W of E assigns to each trivializing patch U C (i.e., patch over which E is trivial) components tv such that in an overlap :
Mil
Mil
= Cv v t v A vector-valued p-form, in Section 9 .3, associated to each p-tuple of tangent vectors to another element of the same tangent bundle over the same point. An E-valued p-form will associate to each p-tuple of tangent vectors V I , . . . , v to M at x E M an element of the bundle E over x . tv
M
TM
p
429
CONNECTIONS IN A VECTOR BUNDLE
n E ·valued p.form \lJ assigns to each trivializing patch U A ordinary exterior p-form 1/!u , such that in an overlap we have
1/!v
C
Mn
a K -tuple of
= C v u 1/!u
For example, if a P is a globally defined p-form a (9 W defines a p-form section of E, th en \}J : == P
1/!U ( V l , " " vp )
==
( 1 6. 1 3)
on Mn , and if W is a global section of E by
a P ( v l , . . . , vp ) W
A connection V for E is an operator taking sections \}i of E into V'l1 such that the Leibniz rule holds; if f is a function, then
V ( \}i f) = ( V \}i ) f + \}i
0
Let e == (e j , . . . , e ) be a frame of sections of E over K Vea i s an E-valued I -form, and thus is of the form
E - valued
I -forms
( 1 6. 1 4)
df
the trivializing patch U . Then
Ve = e @ w or
( 1 6. 1 5)
where
is some K x K matrix of I -forms on U . (We shall try to use consistently Greek letters a, p, and s o on, or Roman capitals for fiber indices 1 , . . . , K and Roman lowercase i. j . . . for Mn indices 1 , . . . , n.) We shall also frequently omit the tensor product sign. Note that the connection coefficients w�.8 have a mixture offiber and manifold indices. Here we are assuming that (X i ) are local coordinates for U e M. For a section W = e 1/! , we have by Leibniz
that is,
V (W)
==
V ( ea 1/! a ) = V ( e1/! ) =
Vw
=
V ( e 1/! )
=
( V e) 1/!
e (9
+ e(d 1/! )
V 1/!
where
( 1 6. 16) In full.
V 1/! a The
=
d 1/! a
+ wa .8 1/! .8
boldfaced V operates on sections, whereas V over the patch U .
of sections
==
V u operates on t he components
430
VECTOR BUNDLES IN GEOMETRY AND PHYS ICS
Suppose now that W is a section over U n V . In order that V be well defin ed. \\Ie require in U n V what physicists call covariance, that is, 1/rv
=
cvu 1/ru
=>
Vv 1/rv
cvu Vu 1/ru
=
( 1 6. 17)
where C vu is the K x K matrix C vu = ( ce u u ) in GI( K ) . As in Section (9. 4) this requires ( 1 6. 1 8)
Note that in our conventions 1/rv
=
ev
=
cvu 1/ru -I
e uc vu
( 1 6. 19 )
=
e u cu v
As usual, we define the covariant derivative respect to the tangent vector X on Mn by
Vx w
of the section
e V lJI ) (X) = (e 0 V 1/r) (X)
Vx eW) : =
=
q, of E with ( 1 6.20)
e[ V 1/r (X) ]
Thus VxlJl = =
e[d 1/r + w1/r] (X)
e[X(1/r) + w (X) 1/r]
( 16.21 )
where Then We then write V x lJl
=
e Vx 1/r Vx 1/r = X i Vi 1/r
where n
,"u
V i 'l'
_ -
O ,I,U 'I'
( 16.22)
u ,1, fJ . + W'fJ ' 'I'
--
ox '
We have defined the covariant differential on sections of E (in a sense, on o-forms whose values are in E). As in Section 9.3d, we now let V send E-valued p forms into E valued (p + 1 ) -forms by defining the exterior covariant differential (again denoted -
by
V)
( 16.23)
where, as in 9.3d, we write 1\ rather than 0 1\ .
431
CONNECTIONS IN A VECTOR BUNDLE
curvature is
introduced as before
e0
V 2 ( e)
=
V ee 0 w)
e
=
dw + w !\ w
e a fJ
-
d wa fJ + w'" y !\ w y fJ
where
=
e
( 16.24) -
1
2" R'" fJ ij d X i !\ d X
j
Note the mixture of indices in the curvature tensor. There i s no notion of torsion in a connection for a general vector bundle. As a simple example, consider the normal 2-plane bundle to a curve M ' , x = x(t) in 1R3 . If v vet) is normal to M along M, we wish Vv = (Vv /dt)dt to be a normal I -form on M. Let d be the usual differential operator for ]R3 ; it is the t ed vec o r valu covari ant differential for the tangent bundle for ]R3 with the usual euclidean flat metric. We shou ld then put Vv : = dv - (dv, T) T ( 1 6.25) where T is the unit tangent to M. For a local description, let u, and U 2 be two normal vector fields along M that are orthonormal. Then the prescription ( 1 6.25) translates to =
Vv = (dv, u , ) u , + (dv, U2 ) U2 In particular, since du, is orthogonal to u" Vu, = Ua Wa , = (du " U 2 ) U2 , shows that w' , = 0 and w2 , = (du" U2 ) = (du, /dt , u2 )dt . When t = s is arc length along the curve M, and when u , is chosen to be the principal normal u to the curve, then, as in Problem 7. 1 (2), U 2 = T x u , is the binormal B, and - (du , /ds , U 2 ) is the torsion r of the space curve. Thus V u = -Br (s)ds and V B = ur (s)ds . 16.3b. Complex Vector Spaces Quantum mechanics deals almost exclusively with complex wave functions and K component wave functions, in other words, with sections of complex vector bundles. (We shall consider quantum mechanics in Section 1 6.4.) Consider the complex plane C with coordinate z = X + iy. C is a I -dimensional vector sp ac e because we allow complex scalars, but C can also be thought of as a real
2-dimensional vector space ]R2 ,
z = x + iy �
[;]
and addition of complex numbers corresponds to vector addition thing about C is that it has a fascinating product Of co urse,
Z , Z2
=
(x , + iy d (X2 + i Y2 )
=
(X , X 2 - y , Y2 ) + i (X ' Y2 + X2 Y ' )
[][] [
this can be expressed entirely in real terms X' y,
0
X2 Y2
in ]R2 . The interesting
=
X ' X 2 - Y ' Y2 x , Y2 + X 2 Y '
]
432
VECTOR BUNDLES IN GEOMETRY AND PHYSICS
In part�cular, multiplication in C by the unit i translates in real terms to a linear � fu��oo whose matrix is
[ � -�l
J= /.
with, naturally, J2 = - Similarly, complex K space, C K , the vector Space (with complex scalars) of complex K -tuples Z = (Zl , . . . , ZK l (Xl + iYl , . . . , XK + i Y K ) T can be considered as ]R2 K under the identification Z ¢> (X l , Y l , X2 , Y2 , . . . , XK , YK ) T = x and then multiplication by i in C K , Z f--+ i z , is translated into a linear transformation J : ]R2 K ---+ ]R2 K with matrix 0 -1 =
0
0
-1
0
J=
0
( 1 6.26)
0 0
-1
0
again with J 2 = - / . Note that J : ]R2 K ---+ ]R2 K is an isometry with respect to the usual metric (x, x')
=
( J x, JX')
since it merely rotates each coordinate plane Yet through 9 0 degrees. Now let F 2 K be any real even-dimensional vector space with an inner product (, ) and let x,, ,
J : F ---+ F
be any linear isometry of F (orthogonal transformation) that is also an anti-involution. that is, J2
= _/
Clearly the eigenvalues of J are ±i , and so det J = 1 . Thus J E S O (2k) an d assumes the form ( 1 5 .6) in suitable orthonormal coordinates ( y" X2 . Y2 , . . . , Xk o yd . But J is skew symmetric, x"
( lx, x')
=
( J2 x, J X') = (x, - J x')
433
CONNECTIONS IN A VECTOR BUNDLE
tells us that in these coordinates J has matrix ( 1 6.26), since each ek must one can introduce complex coordinates in F by putting z" = x" + i Ya , and F then corresponds to multiplication by i . In p arti cu lar, JR 2 with J as earlier can be considered a complex I -dimensional vector C' C, which can be called a complex line. spaCe
atio n 1 5 .6 Equ Then
�:1fF
�
==
16.3c. The Structure Group of a Bundle In a ve ctor bundle each Cu v (x) E GI(n) . We have seen that for a Riemannian manifold !tin, we may choose Cu v (x) E 0 (n) by using orthonormal frames. In a general bundle,
it may be pos
sible to choose the Cu v (x) such that they all lie in a specific Lie group G Cu v : u n V ---+ G We then say that G is the structure group of the bundle. 2 Let M be an oriented Riemannian surface. We can cover M by patches U, V, . . . each of which supports a positively oriented orthonormal frame {eu } , {ev } , . . . of tan gent vectors. Suppressing the patch index, eu is a
=
(e" e2)
positively oriented orthonormal frame in U. It is then clear that each transition for E = TM is a rotation matrix cos a (x) - Sin a (X) c u v (x) = E S O ( 2) cos a (x) s m a (x ) We may say that the orthonormal frames have allowed us to reduce the structure group from GI(2, JR) to S O (2). matrix
[
]
.
16.3d. Complex Line Bundles
Define J acting on the tangent planes of an oriented surface, to be rotation through a right angle in the positive sense; thus
J
:
M; ---+ M; , simply ( 1 6 .27)
and of
Je2
=
-e,
course J 2 = - I . It is clear that J is globally defined; in an overlap U n V the action of J using the frame ev coincides with the action of J using eu . Thus J allows us to consider each fiber in T M 2 as a complex line! The real vector e, E MP2 JR2 �
can be
considered as a complex basis vector e : = e, E M� of the complex line M; . Then
�
C'
434
VECTOR BUNDLES IN GEOMETRY AND PHYSICS
In terms of these bases e U = ef , e V = e i , . . . , the previous SO (2) transition matri ces, c u v (p) , become simply the complex numbers C U V (p) = e i ct ( p )
Figure 1 6. 1 3 The tangent bundle to a n oriented Riemannian suiface can b e considered a s a complex line bundle! The structure group of this bundle is now U ( l ), the unita ry g roup in 1 variable!
The Riemannian connection for M 2 is a connection for the real 2-dimensional tangent bundle. In terms of the orthonormal frames e U , e v , . . . , we have Vei = ej
and we also know Wij
=
-Wj i ;
j
18) W i = ej 18) Wj i
thus
Ve l = e2 18) W2 1
A connection matrix for a complex line bundle would be a I x 1 C
( 1 6.28)
matrix, that is, a single ( for complex). We should then have, in our line
I -form, which we shall denote by W bundle version of T M 2 (where e = e d C
and since e2 = i e l we can rewrite ( 1 6.28) as vee
=
Vel = i e l
18) W2 1 = e l 18) i W2 1
or
( 1 6.29)
W
c
:
·
= l W2 1
=
.
-I WI2
Does this mean that WC = - i W 1 2 defines a connection for this complex line bundle version of T M 2 ? For this to be true we certainly must have that VC commutes with multiplication by complex constants for any cross section 'Ij; (i.e., any vector field on M 2 ) . For example VC (ie)
=
Ve2
= i (e
=
el
0 WI2
=
iel
0
(-iwl2)
18) we") = i Vce
as desired. This connection will be discussed further in Problem 1 8.2(2).
THE ELECTROMAGNETIC CONNECTION
What i s the
435
curvature for this complex line bundle connection? It is the single 2-form e e = dwc + w e
/\
wC
=
dw e
or
( 1 6.30)
where ag ain
K is the Gauss-Riemann curvature R I2 12 of M 2 . P roblem
-
16.3(1 ) If V and
V' are connections for bund les connection for E 0 E' can be given by V"x (
for local sections
=
0 lJI)
=
ea ¢ a and
(Vx
lJI
=
E and E' respectively ove r M then a 0 lJI +
e' w R . Show that for A = e a R
0 e'R).. a R
a ).. a R) + wa ).. b R + w, R'. S ).. a s 'V " ,.().. a R ) _ - ,( jb
1 6.4. The Electromagnetic Connection What does the electromagnetic field have to do with parallel displacement of a wave function?
In Sec ti on
16.4a. Lagrange's Equations without Electromagnetism
10.2a we showed that Lagrange's equations for a massive particle, dp jdt = with p = a L j a i] , follow from Newton's equations Vi] jdt = - grad V . Al though both sides of Newton's equations are contravariant vectors along the extremal q q (t), it is not true that both sides of Lagrange's equations are covectors along the ex tremal, since dp j dt is an ordinary derivative (rather than a covariant derivative) and also aLjaq, =
{
}
aL l agij (q ) . ; . J a v _ qq - k aqk "2 � aq -
( 1 6.3 1 )
is not a covector field because of the first term. To remedy this we may consider the covariant derivative of the momentum covector
that is, This
{ }-
V p; V i]J } Vi]J _ {g ij _ g'.j. dt dt dt Vp dt
av aq
_ _ . . Jk a v g'j g aqk
( 1 6.32)
is a geometric version of Lagrange's equations; the left side differs from dpjdt used; the right side uses the potential function V rather than Lagrangi an L .
in th at a covariant derivative is
436
VECTOR BUNDLES I N GEOMETRY AND PHYS ICS
Let us verify that ( 1 6.32) really reproduces Lagrange's equ ation s, by co . . i , th at IS, mpUtino wh at rJ s ays
dPi Idt - Pj r]kiq· k - a Vlaq' dPi Idt - P i ki q k - - a via q dPi g r q r � gjs{ a gSk ag'ki _ agki } q. k _ av dt ] 2 a q i a q a q S a q i �2 q.s q.k{ aagski + aag'ki aagkSi } aavi q q q q � q s q k{ ��: } �� :� .
.
.
.
0
+
.
.
=
,
_
= =
_
=
_
from ( 1 6.3 1 ) . Combining this with
Pi - giJ q j - aaq i { 2"I grsq. r q. s} - aaTqi - aaLqi _
_
.
_
_
then yields Lagrange's equations, as promised. It is important that
avl a q
==
O.
16.4b. The Modified Lagrangian and Hamiltonian
Consider a charged particle moving in an M 3 with no external electromagnetic field V be the Lagrangian . The particle then obeys la present. Let L L (x , i ) grange's equations = where : = is the kinematical momen tum, that is, the covariant version of the velocity. Suppose now that an electromagnetic field is present also. The particle then suffers but also an additional Lorentz force whose con not only the original force travariant version is e(E + v x B) . This additional force is not the gradient of a potential and so we cannot get the complete Lagrangian equations of motion merely by adding a new potential term V' (although we could if the magnetic field were not present). It turns out, though, that we can write the equations in Lagrangian and Hamiltonian forms if we make a more sophisticated change. For this purpose we shall consider a massive charged particle, moving perhaps relativistically in ffi. 3 . We shall first make some heuristic remarks (inspired by comments of Weyl [Wy. pp. 52, 99]) concerning the notion of the Lagrangian in particle mechanics and the changes when an electromagnetic field is present. Unlike the total energy T + V , which is frequently a constant of the motion, the Lagrangian - V seemingly was introduced merely to make Lagrange's equations take a simpler mathematical form. Although introduced long ago, I feel that its physical significance could not be appreciated before the introduction of special relativity. Introduce units for which the speed oflight is unity, c = 1 . Special relativity associates to the world line of a massive particle its energy-momentum 4-vector P = (E, p l. E m + (1 /2)111 0 v 2 + , which, except for a constant m o , reduces to the classical kinetic energy T for low speeds. If the classical force is derivable from a classical backround potential V , t V , special relativity suggests that we should augment the energy E by V , yielding a "total energy" H E+V 111 0 + (T + V). as in section 7. I c. We may then form, as a first attempt, the "total energy momen� 4-vector" (H, Put Ho H - 111 0 + V . The I -form assoc iated to ( H , p) , =
Tdpld t aLlax, =
P aLlai
- av lax
T
=
�
.
111 0
.
.
=
p)T.
-
:=
'V
:=
�
T
�
437
THE ELECTROMAGNETIC CONNECTION
.
I.e.,
the
total energy momentum covector is then Pa dx a - H dt
=
Pa dx a - Hadt - m o dt
bich
� is
i s the extended Poincare I-form or action I -form (4.57) augmented by a term lfIodt which does not alter the equations of motion. Along the world line, this I -form
[Pa ( d:ta ) - Ho] dt - modt
=
Ldt - m odt
tbe Lagrangian action integrand Ldt is, except for a disposable exact differential, me total energy-momentum 1-form, in the sense of special relativity, along the world
\iDe! This, I believe, explains the significance of the Lagrangian in the principle of least
action !
There i s a disquieting feature of the above argument; we took a 4-covector Pa dx a Edt and added to its time component - E a scalar - V . This violates "Lorentz covari ance"; one cann ot add a scalar to one component of a covector. (This does not mean that the above procedure is invalid; it makes perfectly good sense if we agree to use only those Lorentz transformations that do not involve time, for example the usual changes of spati al coordinates traditionally used in non-relativistic mechanics.) The situation is much more satisfactory when the backround field is the electromag netic fi eld with covector potential A = ¢dt + A a dx a or vector potential ( ¢ A) T . In thi s case V = e¢ can be added to E provided we add eA to p, since it makes Lorentz sense to add two 4-vectors together! The resulting covector, the total energy momentum I-form is simply
,
with Lagrangian
Ldt This
-
-
-
=
,
[CPa + eA a ) ( d:ta ) - (E - e¢) ] dt - modt
also suggests that if one has a classical dynamical system, with Hamiltonian H and
no electromagnetism, then to get the Hamiltonian equations when electromagnetism is introduced one simply defines a new Hamiltonian by H * := H - e¢ and new momenta
by P*a and th
:=
Pa + eAct • But then Pa* dx a - H * dt
=
Pa dx a - Hdt + e(A a dx a + ¢dt)
e extended Poincare 2-form should be redefined to be Q*
:=
d(Pa* dx a - H * dt)
=
Q + eF
( 1 6.33)
(It can
then be shown that Hamilton's equations are now i x Q * 0, where X = (dx /dt) a/ax + (dp /dt) a lap + a/at uses the original P rather than the augmented p * .)
We are now finished with our heuristic discussion and we proceed with our formal veri fication of these hopes .
438
VECTOR BUNDLES I N GEOMETRY AND PHYSICS
Theorem (16.34): Let H = H(q , p, t) be the Hamiltonian for a cha rged ticle when no electromagnetic field is present. Let an electromagn etic fie introduced. Define a new canonical momentum variable p* in T* M x lR by
:�
p\,
:=
Pa + eA a (t , q )
and a new Hamiltonian H * (q , p * , t)
H (q , p * - e A , t) - e ¢ (t, q ) Then the particle of charge e satisfies new Hamiltonian equations dp* aH* dq a H* dH* a H* and = and = dt dt ap * aq dt at PROOF :
:=
H (q , p, t) - e (t , q)
=
Compare the solutions of the original system dq dt
aH ap
dp dt
and
-
-
aH aq
- -
and the new system dq a H* dp* a H* and = dt dt ap * aq At a point (q , p , t) = (q , p* - eA, t) we have a H * a H (q , p* - eA) a H (q , p) dq = ap * ap * ap dt and so the velocities dq / dt are identical in both systems. Denote - a H/aq by f, the force in the original system. Then dA dPa a H a H a (-eA,B) e!.t dpa* a H * + = e a + a + a a dt dt aq dt aq ap f3 aq aq a dA dPa dq f3 a A f3 - e !!t = + e a _ fa _ e dt dt dt aq a aqa -
( + ) ( )
=
(v
x
dA aA B) a + -a - a dt at
=
_
-
Thus dpa* a H * -- + --a dt aq
[ ] ( )( )
[
dpa - fa - e (v dt -
dpa
= dt
- fa - e[(v
x
X
a aA B) a - -a + -a at aq B)
+
EJ a
]
439
THE ELECTROMAGNETIC CONNECTION
lIenee dp* Id t = the Lorentz force,
- a H * laq is equivalent to the original system augmented by D
as desired.
The Lagrangian
Along
a lifte d
L and the Hamiltonian H are related by L (q , q) curve q = dq Idt we then have
In termS of our
L (q , q )dt
=
=
pq - H (q , p).
pdq - H (q , p)dt
new Hamiltonian, we should define
L * (q , q )dt
:
= =
=
p * dq - H * (q , p * )dt ( Pa + eA a )dq a - [H (q , p) - e¢]dt [Pa dq a - H (q , p)dt] + e[¢dt + A a dq a ]
( 1 6.35)
Corollary (16.36): A particle in an electromagnetic field satisfies Lagrange 's equa tions a L * l aq - dldt (a L * laq) = 0 with new Lagrangian L * (q , q) = L (q , q) + e[¢ + A a q a ] that is,
L * dt
=
Ldt + e[¢dt + ti]
=
Ldt + eA I
16.4c. Schrodinger's Equation in an Electromagnetic Field
In the present section we shall remove the mass termfrom the metric; energy of a particle is the familiar
T
1
=
2 m (q , q ) =
that is, the kinetic
p2 2m
Consider a charged particle, of mass m, moving in a potential field V field in ]R3 , with no external electromagnetic field present. If we neglect spin, the electron is commonly represente d
in quantum mechanics by a wave function
a complex-valued
1{r (x ) = 1{r (x, t )
time-dependent function on ]R3 . Schrodinger's equation states that the wave functions evolve in time according to
a in 1{r = H 1{r at where the Hamiltonian operato r H is defined as follows. The Hamiltonian
( 1 6.37)
of a particle in classical mechanics is given by H (x, p)
=
p2 + V ex) 2m
( 1 6 . 3 8)
where p is the canonical momentum. Schr6dinger then postulates that in Cartesian Coordinates the canonical momenta P are represented by the differential operators a
Pa
=
-in
a
ax a
( 1 6.39)
440
VECTOR BUNDLES IN GEOMETRY AND PHYSICS
The potential is simply the multiplicative operator 1/1 comes, in Cartesian coordinates in Il{3 ,
V
( rz2 ) -a21/1a at 2m � (aX ) 2 e eA )2 2m
. a1/l
- - '"
In -
+
f-+
V(x)1/I ,
and ( 1 6.38
V 1/I
) be.
06.40,
If the particle has charge and there is an additional external electromagnetic fie lei present, then ( 1 6. 34) says that ( 1 6.38) should be replaced by (p* + V (x) H (x, p * ) =
-e
0'
and the canonical momenta p* should be replaced, when are Cartesian coordinares, by P*a = n / SchrOdinger's equation becomes 0'
In - =
If we write this in the form -
h
I
L
- In
-
0'
=
n2
-
L
h
0'
we may then write
=
L
(16.4 1 )
+ V1jJ
+
0'
where
=
-
and
h
( 16.42)
=
We may write, instead of the last two definitions,
( 16.43)
We then have the following situation. We originally thought of as being a complex function on Il{4 , that is, a section of the complex line bundle over Il{4 . SchrOdinger's The vector + equation involves the vector potential = A t potential is not uniquely determined; we may, if we wish, use a differen choice Ah in each of several patches U in Il{4 . If we do so, then in each patch U we shall have precisely a different SchrOdinger equation, satisfied by a local solution =
-.
Equation ( 1 6.43) then takes on the appearance of a
V·1/I } axa 1/1} + w} 1/l :=
where
o ver R4!
( 1 6.44)
441
T H E E L E C T RO M A G N E T I C CONNECTION
If i s to be a connection, what is the bundle? Consider two OJ 4 (lVd'lapping p atches of lR In these patches we have .
wu
=
-
C:)
Wv = -
Au ,
C:)
choices A u and A v in
Av
. ee the electromagnetic field 2-form, F = dA, is well defined, it must be that A v - A u closed I-form on U n V , and, if this intersection is simply connected, A - A u is
!I:
v
cUet,
+ dfu v
in U n V "here fuv is a real single-valued function on U n V . Then Av
=
Au
C:)
Wv = W u -
dfu v
But a c onnection i n a bundle transforms b y ( 1 6 . 1 8), and when the bundle is a complex Jine bundle, the Cu v are 1 x 1 complex matrices and the transformation rule becomes
Wv
Thus we may
choose log c u v (x)
=
c u v (x)
=
+
Wu
c u � dc u v
( 1 6.45)
- (ie/h) fu v , that is, =
exp
{-Cr: ) fu v }
( 1 6.46)
ifcuvcvwcwu = 1 is satisfied then ( 1 6.46) defines a line bundle whose cross will be our local wave functions. A wave function is then not a single complex- valuedfunction
tu , 1/Iv ,
. . .
sections
1/r but rather a collection
offunctions such that in an overlap U n V
1/rv (x) = c vu (x) 1/ru (x )
=
exp
{ (�) fu v } 1/ru (x)
( 1 6.47)
This brings us back to the starting point of gauge theories in quantum mechanics, namely
Weyl's principle of gauge invariance (16.48): If1/r satisfies Schrodinger 's equa tion (16.41), which involves the potential A, then satisfies Schrodinger 's equation when A has been replaced by
A + df To see this
says that if
let U = V but let us choose A v = A u + df Then Weyl's principle simply .
442
VECTOR BUNDLES IN GEOMETRY AND PHYSICS
then ( 1 6. 1 7), that is, cvu V u = V V cvu , shows that this same equation holds . V V V u ) when 1fr v is replaced by 1fru ! Note that without the notion of a conn (�th the verification of this would be messy; Schrodinger's equation, when writt�OQ n OUt without covariant derivatives, involves H-
VIL V� 1fr =
;��2 - ( � ) [A � ;! o�� CA� 1fr)] +
-
C 1 6. 1 7) is the crucial simplification.
(::) A�A�1fr.
It should be clear that Weyl's principle is not restricted to Schrodinger 's equati n . o , "covariance," that is, C 1 6 . 1 7), is the essential ingredient. Note that the transition functions ( 1 6.46) for our bundle are complex numbers f absolute value I ; the structure group of the given line bundle is the group U ( 1 ) . n:;s implies that l 1frv l 2 = l 1fru l 2 and consequently the probabili ty interpre tation 0/ 1 1fr 12 in
quantum mechanics can be maintained. The curvature of the connection from C 1 6.44) is essentially the electromagnetic field
2-form.
e = dw
= =
(ie) - ( � ) [ c.�
=
- h dA I /\
( ie )
- h F2
( 1 6.49)
dt + �11]
Finally, we shall make some remarks about Schrodinger's equation in curvilinear coordinates. Consider a Riemannian manifold M, the most important case being R3 with a curvilinear coordinate system. Let E be a complex vector bundle with connection w, for example the wave function bundle with w = - ieAjh . We suppress the bundle indices on w and on 1fr . For covariant derivative we have 1fr/j = Oj 1fr + Wj 1fr. This represents a cross section of the bundle E 0 T* M, that is, the bundle of covariant vectors on M whose values are in E . As we have seen in Problem 1 6 3(1 ) the covariant derivative 1frm = C1fr/j )/k of this tensor will involve not only the connection for E but also the Riemannian connection r .
,
W
1frm = C1frj ) / k = ok 1fr/j =
+
Wk 1fr/j - rk rj 1fr/r
Ok COj 1fr + Wj 1fr) + Wk COj 1fr + Wj 1fr) - rk rj COr 1fr + Wr 1fr)
= [Ok Oj 1fr - r/j Or 1fr] + {Ok CWj 1fr) + Wk COj 1fr + Wj 1fr) - rk rj Wr1frl which is now a covariant second rank tensor on M with values in E . Then
"Laplacian"
the
n 2 ,/, .. _ g j k ,/r/jk v
-
'P
'P
is again simply a section of E . In slightly more detail
g j k [Ok Oj 1fr - rkj Or 1fr ] + g j k { l The term involving the square brackets g j k [ ] is simply the Laplacian of the "function" 1fr, using the Riemannian connection 1 0 0 1fr gg j k .jg o xj .j ox k V 2 1fr
=
g j k 1fr/jk
=
[
( )]
443
THE ELECTROMAGNETIC CONNECTION
ion (1 1 .30)). A candidate then for Schrodinger's equation for a charged equa t �cle n electromagnetic field on M would2 be l.h ( aat - Tie
jk
SOJIlIllary. When no electromagnetic field i s present, the Hamiltonian i s o f the form.
I
2m
in
La g':tfJ pa PfJ
+
V
coordinates or on a Riemannian manifold M. We replace Pa by the c i inearcovariant derivative - i h V:: . SchrOdinger's equation becomes a 1/1 = (- -h 2 ) --I --a a [JggafJ ( a 1/1fJ ) + V 1/I .lhat 2m g ax ax urv
l
Riem an nian
J
The only effect of introducing an electromagnetic field now
J
h, and we must use the
is to replace the trivial wave
function bundle by the bundle E with connection w = - i e A / full c varian deri vati ve (using both r and w)
t
o
h / (- ;� ) gafJ 1/I/afJ + V1/I
i 1/l o
=
In this procedure there is no need to first introduce the the classical system augmented by the electromagnetic
In m
new canonical momenta p * in field!
16.4d. Global Potentials
o t problems involving electromagnetics the vector potential I-form
Al
is glob ally defined. We can see this as follows. Consider first a smooth electromagnetic field F2 in all of Minkowski space, Mri. Since Mri = ffi.4 has second Betti num ber b2 0, de Rham's theorem assures us that there is a potential I -form A 1 for the closed 2-form = d A l . Usually, however, there are singularities for located, for example, at the moving point charges. We cannot apply de Rham's the orem to singular forms; thus in order to use de Rham's theorem we must first re move the singularities of from Mri, leaving an open subset U of Mri. Now, how ever, there is no reason to assume that b ( U ) = 0; for example, a fixed charge at the origin of ffi. 3 yields an entire t axis of singularities in Mri, and the 2-sphere in 13 surrounding the origin is a 2-cycle of U = Mri - (the t axis) that does not bound. In spite of the fact that U may have nonbounding 2-cycles, we still have the s
F2 , F2
=
F2
F2,
2
fOllowing:
space-time M 4 x ffi. with V 3 a spacelike hypersurface and t a global coordinate for ffi.. Suppose that the mag netic field has a globally defined potential vanishes at time t = O. Then A l , d A l = on all of U.
Theorem (16.50): Consider a region U of a general relativistic that has a global time coordinate, that is, U is of the form v 3
�J32 2
F,
F2
444
VECTOR BUNDLES IN GEOMETRY AND PHYSICS
F 2 is closed, d F = O. By de Rham's theorem, we need only s how that the integral of F 2 over each 2-cycle z of U vanishes. But z can be deformed , by the deformation cjJcx (x, t) = (x, ( 1 - a)t), into a homologous spatial cycle z' that lies in the hypersurface where t = O. Since F 2 = t:� 1 1\ d t �R2 restricts to 0 on the deformed cycle, Jz F 2 = Jz' F 2 = O. 0
PROOF :
+
For a simpler discussion in ]R 3 , see Problem 1 6.4( 1 ). In the standard cosmological models, the Friedmann universes, there is a global time coordinate (see [F, chap. 1 2]). Thus the only way F 2 can avoid having a global potential today in these models is for there to have existed, since the time of the big bang, a nonbounding 2-cycle, and a magnetic field with nonzero flux through this cycle. (Some of the Friedmann models do have b2 =I=- 0 ; for example, there are mOdels where the spatial sections V 3 are flat 3-dimensional tori T3, and others with V 3 clOSed manifolds with negative curvature and b2 =I=- 0.) 16.4e. The Dirac Monopole
If there is a global potential A I , there is then no necessity for introducing a bundle whose sections will serve as local wave functions, since one global patch U will suffi ce . It may very well be though that when considering other fields, for example the Yang Mills fields, to be discussed later, we shall not be so fortunate, and in that case we shall be forced to introduce bundles and connections, as we have had to do in the case of gravitation in general relativity. There is, however, a much simpler situation that requires bundles, namely the Dirac magnetic monopole (which, however, has never been shown to exist). Consider then an electron moving in ]R3 - { O } in the field of a magnetic monopole of strength q fixed at the origin. The B field for this monopole is B = (q/r 2 )8/8r, that is (see equation (5 .9)), �1\ 2 = i vol3 = d[q ( 1 - cos e)dcjJ] Thus (:f� = q ( 1 cos e)dcjJ in the region U = ]R3 - {negative z axis} . We shall need also to consider points on the negative z axis (except for the origin). In the region V = ]R3 - {positive z axis} we can use e' = e and cjJ' = cjJ as coordinates and get (:/� = -q ( 1 + cos e)dcjJ Maxwell's equations hold everywhere on ]R3 - {OJ. Since a� does not agree with (f� in U n V = ]R3 - {z axis}, we shall be forced to introduce the electromagnetic bundle and connection of Section 1 6.4c. In Problem 1 6.4(2) you are asked to show that the transition function for this monopole bundle is B
7T
-
-
-
C vu
=
(
2ieqcjJ
exp - -h-
)
( 16.5 1)
445
THE ELECTROMAGNETIC CONNECTION
not single-valued unless Dirac 's quantization condition
ha is Note t t this
2eq
h
's ati sfi ed .
must be an integer
( 1 6.52)
If this condition is not satisfied we shall have failed in our attempt to Since there are only two patches U and V , Equation ( 1 6.3) is � Thus if ( 1 6.5 I ) holds, the monopole bundle will exist. i s not in general single-valued is a reflection of the fact that in this case Th at Cv u 3 u n V = ]R {entire z axis} is certainly not simply connected (more to the point, its does not vanish). It is true that by using more sets (whose intersections number tti t Be firs s to cover lR 3 - {O}, we could find transition functions that would connected) mply are i be si ng le-valued without requiring ( 1 6.5 2 ) , but it would tum out that it would not be possible to satisfy the crucial equation C u v C v w C w u = I . In fact, we shall prove in Section 1 7 . 4, from a general Gauss-Bonnet theorem, that for any complex line bundle over ]R 3 -o rigin, the curvature must satisfy s
ndle . ons tru ct a bu auto maticall y sati sfied.
-
i __ 2;rr
r
i __ eZ 2;rr 1M2
in the geometrical
oriented closed
( 1 6.49) ,
integer
=
- {O} ,
Z) .
integer
=
ie �J.l n
--
=
ie n
. vol3
- -IB
=
-
( 2i ) ( i eq ) (4;rr ) ;rr
h
. r
i eq 3 l a / a vol r Zn
- -
thus the integral i n ( 1 6.53)) becomes
Thus,
( 1 6.53)
case when the complex line bundle is the tangent bundle to the surface MZ; see (9.66) and ( 1 6.30) ! ) For the monopole bundle, from e
and
=
S2
S Z i s a generator for the second homology group HZ (lR3 have already proved
The unit sphere (Note that we
J eZ
=
2eq
h
as noted first, I believe, by Sniatycki [Sy]
ifDirac 's condition ( 1 6. 52) is not satisfied, there will be no complex line bundle whose sections can serve as wave "functions " for the electron in the field of a magnetic monopole. (For a description
of the monopole bundle, see Section 1 7.4c.) This yields a quantiza
tion condition, relating the charge on a monopole to that of the electrons.
it will be shown in Chapter 1 7 that the flux of e�B /2;rr n through any surface, for any magnetic field, must be an integer.
More generally, closed oriented
446
VECTOR BUNDLES IN GEOMETRY AND PHYSICS
Figure 1 6. 1 4 z'
Figure 1 6. 1 5
Z2
In Figure 1 6. 1 4 we have indicated a V3 that consists of two separated horizontal sheets (two "separate" universes) that are joined by a wormhole cylinder S2 [0, 1]; we have indicated one of the spherical sections going around the "throat" of the wonnhole. A magnetic field goes from the bottom sheet through the lower "mouth," threads through the throat, and comes out of the top mouth. In this example Jz �i\2 f:. o. Figure 1 6. 1 5 is similar except that the wormhole joins two distant portions of the "same" universe, and again B has a nonzero flux through the throat. In both cases there is no global A and the flux of B through the throat must be quantized in terms of e . Finally, we wish to emphasize one point. Ifthere i s a monopole, then from ( 1 6.5 1 ) we see that o/ v = exp( - 2i eq¢ Iii ) o/u #- o/u . Thus the electron wave function 0/ cannot be defined (and single-valued) everywhere and it must, rather, be considered as a section of the monopole bundle with at least two patches. Presumably, then, we should expect that other types of fields that interact with elementary particles might demand that wave functions be replaced by sections of bundles, just as we do not expect that every manifold should be covered by a single coordinate patch. x
16.4f. The Aharonov-Bohm Effect
At first sight the electromagnetic connection = - i e A I Iii seems nonphysical since classically the vector potential A I , changing with each choice of gauge , was regarded only as a mathematical tool for describing the physical electromagnetic fi eld F2 :: w
447
THE ELECTROMAGNETIC CONNECTION
A I . We h ave
noted, however, a similar situation in general relativity; the Levi-Civita nnection a gauge field, can be thought of as merely a preliminary mathematical step CO the way to its derivative, the Riemann curvature tensor, describing the strength of gravitation�l field. How ever, �his gravitational connecti � n is a physical field in the . . governs parallel displacement. We shall now et it, wIth no use of Its denvatlves, n s h at I th at the electromagnetic connection, that is, the vector potential A , although not a c ass ic al physical field, is a physical field in quantum mechanics, and this will be l illu strated with the famous Aharonov-Bohm effect. Wi th a sole noid carrying a current j, the circulation of the magnetic field about a e C going around the coil is, by Ampere's law, fc �g 4][j . When j is very clos d l oop a wire is tightly wound, the magnetic field inside the coil can be substantial the , if sm ll is very small. In the simplified version of an infinitely long, infinitely outside fB whi le * h solenoid, it is assumed that the magnetic field inside is constant and wound ti g tl y paral l el to the axis of the coil, and the magnetic field outside the coil is vanishingly J
,
:e :e
*
=
small.
[3 = 0
Looking down a long solenoid
Figure 1 6. 1 6 Let
the magnetic flux inside the coil be JJ �g b. Then (i l = bd8/2][ is a well defined vector potential in the region exterior to the coil, designed to satisfy both Ie (:/ b and dct O. See Problem 1 6.4(4) for the potential inside the coil. I t i s possible to detect the effect of ( 1 on an electron constrained to the exterior reg i on even though B 0 in this region; this is the Aharonov-Bohm effect. A brief explanation in terms of path integrals is as follows. (We assume here a slight familiarity wi th Feynman's method. For more details the reader is referred to Feynman's lectures [F, L, S, vols. II and III], Rabin's article [R ] , and the excellent book [Fe] by Felsager. For insight into the path integral formalism (without mentioning integrals !) see Feynman's remarkable book [F] .) A n electron is emitted from a source, passes through one of two slits in a screen, mov es along a curve y, and strikes a screen behind the solenoid at a point y. The " probability amplitude" for this process is proportional to the exponential of the classical action for the path y =
=
=
=
(� i Ldt )
( 16.54)
principal contribution to the amplitude for going from x to y is given by this when the path y is a classical path of "least" action. Since the electromagnetic field vanishes in this exterior region, the classical path will be a straight line from the
expres sion
448
VECTOR BUNDLES IN GEOMETRY AND PHYSICS
slit i n the screen. We exhibit the two classical paths C and C' (both must b e taken into account since we don 't know which slit the electron chooses).
solenoid
screen
Figure 1 6. 1 7
The phase o f the complex number i s its angle or argument. The phase difference, due to paths C and C', is responsible for the interference pattern observed at the screen. Look at the cases when there is no current in the coil and when the current is flowing. In the first case the Lagrangian is L o . After the current is turned on, there is a vector potential (f present outside the solenoid. Corollary ( 1 6.36) then tells us to replace L = L o by L = L o + e (i; "' A ", + ¢ ) ; that is, we replace Ldt by Ldt + eA . dx, since the scalar potential vanishes. In this new situation the phase difference becomes
and this differs from the original phase difference only by
� JJ �H e; =
Since this is independent ofy for any pair C, C' of classically extremal paths, Aharonov and Bohm concluded that the original interference pattern will simply be shifted by a constant amount, in spite of the fact that the electron feels no magnetic force in the exterior region ! This shift has actually been observed. The field A , and thus the connection w, are physical fields in quantum mechanics.
Problems
1 6.4(1 ) Let Z2 be a closed surface in
------
IR 3 that l ies outside the singularities of the electro
magnetic field . Show di rectly from Maxwell's equations that Jz �g2 is constant in time. Th is shows that if �il2 van ished sometime in the past , then B curl A i n the nonsingular set. =
1 6.4(2) Derive ( 1 6.5 1 ) .
1 6.4(3) I n the monopole bundle with 2eq/h a n integer -=Icross section over U = IR 3 cross section over all of IR 3 negative z axis.)
-
-
0 , the function O/ U 1 i s a (negative z axis ) . Can O/U be extended to be a {OJ? (Look at the proposed O/ V at points of the =
449
THE ELECTROMAGNETIC CONNECTION
1 6 .4(4) Assume a constant axial magnetic field Bdx /\ dy inside the coil whose axis is
the z axis. (Thus B = b/n a2 , where a is the radius of the coil . ) Of course Bx dy is a covector potential, but to match up with our external potential use cylind rical coordinates r, e, z , and show that another choice of a covector potential 1 -form is given by A1 = ( B/2) r2 de . What is the length II A II of this choice for A? What is the length of the exterior version for A used in the text? Why don't they match up smoothly?
1 6 . 4(5) Show the gauge i nvariance of Feyn man's prescription ( 1 6. 54) as follows : Let
a particle in an electromagnetic field have probability amplitude 1/f u (X, O) of being at x E U at time o. Then the probabi lity ampl itude that the particle will traverse a path y from x to y, arriving at y at time t, is, i n Feyn man 's view 1/fu(x, 0) exp
[� 1
Ldt + eAu
•
dX
]
=
1/fu(X, 0) exp
[� 1 ]
Ldt exp
There is a similar expression if we use a different gauge 1/f v(x, 1/fu (x, 0) . Show that these two gauges yield compatible results.
0)
l =
-wu c vu (x)
CHAPTER
17
Fiber Bundles, Gauss-Bonnet, and Topological Quantization A vector bundle is a family of vector spaces parameterized by points in the base space. How
do we parameterize a family of manifolds, say Lie groups ?
1 7 . 1 . Fiber Bundles and Principal Bundles
17.1a. Fiber Bundles The tangent bundle T Mil to a Riemannian manifold is a vector bundle associated to M ; it is locally o f the form U x ffi." . We have had occasion also to consider the set o f unit vectors tangent to M; that is, we may consider, in each fiber n - 1 (p) � ffi.n of TM (a vector space with scalar product), the unit sphere sn- l (p) C n - 1 (p) . The collection of all these unit spheres S"- l (p) , as p ranges over M, forms a new manifold, called the unit tangent bundle To M in Section 2.2b. We again have a projection n : To M -+ M. The term bundle refers to the fact that the space is again locally a product in the following sense: To M" is the collection of all (n 1 ) spheres S" - l (p) in all of the tangent spaces to M, but there is no natural way to identify points in S(p) with points in S eq ) for distinct points p and q in M . Choose an orthonormal frame e U = (el , " " e ) in a n patch U of M and take a fixed unit sphere S in some euclidean space ffi.n • We may then identify each tangent sphere S ( p ) , at p E U, with the fixed sphere S, by identifying v = (ei v i ) E S,,- I (p ) with s = ( V I , . . . , v" ) E S " - I C ffi.1l ; thus (p, v) is identified with s. We then have a diffeomorphism -
x
S" - I
-+
n-I
(U)
C
To M
exhibiting the local product structure. Of course if we go into another patch V , using a new frame e V , then we shall get a different identification. This space To M is a "fiber bundle," but not a vector bundle, because the fiber S" - I is a manifold that is not a vector space. We may now define this new notion in general. To M is atypical, since it is a subbundle of the vector bundle TM . A fiber bundle consists of the following: There are a manifold Fk (called the fiber), a manifold E (the bundle space) and a manifold Mil (the base space) together with a map n :
E
-+
45 1
Mil
452
FIBER BUNDLES, G AUSS-BONNET, AND TOPOLOGICAL QUANTIZATION
of E onto M . We demand that E is locally a product space in the following sense' There is a covering of Mn by open sets U, V, . . . , such that JT - 1 (U) is diffeomorphi� to U x F; there is a diffeomorphism ( 1 7 . 1)
Yu E F. Then for each p E U the assignment Y E F --+ ¢u (p, y) E JT - 1 (p) is a diffeomorphism; that is, the fiber JT - 1 (p ) Over p is a diffeomorphic copy of the fiber F of the bundle. In an overlap U n V a point e E JT - I (U n V ) will have two representations
with ¢ u (p, Yu )
E
JT - I (p) for each
e
=
¢u (p, Yu ) = ¢ v (p, Yv )
and we demand that
Yv
=
c v u ( p ) [y u ]
( 17 . 2)
where C v u (p) : F --+ F is a diffeomorphism of the fiber. In the case of a vector bundle, F = ]R K or C K , each cv u (p) : ]R K --+ ]R K was a linear transformation, but now, of course, F is a manifold and need not have a linear structure. The set of all diffeomorphisms of a manifold F clearly form a group in the sense of algebra. (It is not a Lie group; e.g., the diffeomorphisms of ]R2 form, in a sense , an infinite dimensional manifold). If all the maps CU I' (p) lie in a subgroup G of the group of all diffeomorphisms of F we say that G is the (structure) group of the fiber bundle. In the case of the unit tangent bundle To M to a Riemannian manifold, by using orthonormal frames as we did earlier, each C v u (p) : sn -I --+ sn - I is the restriction of an orthogonal transformation ]Rn --+ ]Rn to the unit sphere sn- I C ]Rn . Thus by employing orthonormal frames we reduce the structure group of the fiber from the group of all diffeomorphisms of sn- I to the subgroup 0 (n) of orthogonal transformations of the sphere. In the case of the normal real line bundle to the midcircle M 1 of the Mobius band (see Figure 1 6 . 1 ) , we may choose unit sections eu and ev in U and V , respectively. On the two pieces of the intersection u n V we have in one case e u = ev and in the other e u = -ev . -
Thus the structure group of this normal bundle is the 2-element multiplicative group {± I } , which is easily seen to be another version of the additive group !Z2 . Of course we still demand
( 1 7.3)
and cu w 0 Cw v 0 C v u = identity on U n V n w
As in the case of a vector bundle, a fiber bundle over M can be constructed as soon as transition functions satisfying ( 1 7 . 3 ) are prescribed. A (local cross) section is again a map s : U --+ E such that JT 0 s = identity. A section s is simply a collection of maps { s u : U --+ F } such that in U n V we have s u (p) = c u v ( p ) [s v ( p ) ] .
FIBER BUNDLES AND PRINCIPAL BUNDLES
453
:rc
--------�--�- M u
Figure 1 7. 1 In Section 1 7 . 2 we shall see many examples of fiber bundles .
17.1h. Principal Bundles and Frame Bundles Let Mn be Riemannian and let FM be the collection of all orthonormal frames f l ' . . . , fll of vectors at points of M. rr : F M ---+ M assigns to each frame f = (f l ' . . . , fll ) the point p of M at which the frame is located. What is the fiber rr - I (p) over p ? Let e be a given frame at p; then the most general frame f at p is of the form
f = eg
i.e.,
where the matrix [gO: I'l l = g E G = O (n) . Thus after a single frame e at p has been chosen, the fiber rr - I (p) of all orthonormal frames at p can be identified with the structure group G = 0 (n) of orthogonal n x n matrices. The fiber for the frame bundle FM is the Lie group O (n ) . How do we exhibit the local trivialization (product structure)? Let e be an orthonormal frame field on an open set U e M ; for example, we can apply the Gram-Schmidt process to a coordinate frame in a patch. Then a general orthonormal frame f on U is uniquely of the form
f(p)
=
e u (p)g u (p)
( 1 7.4)
Thus the frame f in U is completely described by giving the point p and the matrix g u . The local trivialization assigns to each p
E
U and each g
E
G
---+
rr - I (U)
G the frame
In an overlap, the same frame ( 1 7 .4) will have another representation
f(p)
=
ev (p)g v (p)
where
(1 7 . 5 )
e v (p)
=
e u (p)c u v (p)
454
F I B E R B U N D L E S , G A U S S - B O N N E T , A N D T O P O LO G I C A L Q U A N T I Z A TI ON
O (n) is the transition matrix for the tangent bundle (recal l th at in Tid' for a vector y = e u y u = ev y v , then Yv = c v u Yu ) . Then f = e u (p ) g u (p ) = ev (p ) g v (p) = e u (p ) c u v (p)g v (p )
c u v (p) E G
=
gives
g u (p)
=
c u v (p ) g v (p)
( 1 7. 6)
Thus the diffeomorphism
c u v Cp) : [G
=
O (n )]
--+
G
is simply left translation of G by the (transition) orthogonal matrix C u v (p) ! In general we shall say that a fiber bundle { P,
M , rr , F, G }
is a principal bundle if the fiber F is the same as the group G , and if the transition functions c u v ex) act on F G by left translations. =
The frame bundle FM is the principal bundle associated with the tangent (vector) bundle TM .
IRk ,
By exactly the same procedure, given any vector bundle E � M with fiber we can, by considering frames of k l i ne arly independent local cross sections, construct the associated principal bundle P whose fiber is the structure group of the original vector bundle.
17.1c. Action of the Structure Group on a Principal Bundle The frame bundle has a remarkable property that is not shared with the tangent vector bundle: the structure group G acts in a natural way as a group of transformations on FM. Let g E G be a given matrix and let f = (fl , . . . , fll ) be a frame at p, that is; f is a point in FM. Then we can let g send this point f into the new point g (f) : = fg by the usual ( 1 7 .7)
Note that this assignment is intrinsic: we have not used the local product structure ! There is, however, no natural action of G on the tangent bundle itself. For example, if M 3 is 3-dimensional and if v is a tangent vector at p and if g is a 3 x 3 matrix, what would you like g (v) to be? We cannot assign a column to v without first assigning a basis for M\p) , and assigning a particular basis is very unnatural ! It is because FM is the space of bases that we succeeded in ( 1 7.7). Thi s works for any principal bundle, namely:
Theorem (17.8): The structure group G of a principal bundle P acts "from the right " on P Cf E
P, g E G)
--+
(fg )
E
P
without fixed points when g i- e, and preserves fibers (i. e., rr (fg) = rr Cf) ).
455
FIBER BUNDLES AND PRINCIPAL BUNDLES
PRO O F :
We first define the action locally. Let f E U over which P is trivial open
=
P and let n ( f)
=
p lie in some
n-1 U
Then we can write uniquely
that is, f has the local "coordinate" fu E G. We define fg to be the point with local coordinate fu g ,
( fg) u = fu g
Thi s is in fact coordinate independent, for in an overlap U n v , f would have fv = cvu (p) fu and then
(fg) v
=
fv g
=
cvu (p) fu g = cv u (p) (fg) u
D
We see in this proof that the essential point is that left translations in G (say by Cv u ) commute with right translations (say b y g) . We can use the same notation in a principal bundle that we used in the frame bundle. Over U we may consider the local section eu
eu (p)
:=
where I is the identity matrix in G . Then for any point f E n - I (p) we may write for a unique
fu
E
f =
=
eu (p) fu
0 7 . 9)
G.
Each right action f ---+ fg i s a diffeomorphism : P ---+ P . Let the I -parameter subgroup etA , A E �/ , act. The resulting velocity vector field on P is then
A * (f) := In terms
:t [fet A ]t =o
of the local product structure, f = e u fu , and then
A * (f) The action f ---+ fel A on
P
=
d [ e .. e t A = dt U J u ]1 0
is completely described by the action in G
fu
---+
fu et A
whose velocity vector at gu E G is
d et A o dt [fu ]t=
=
fu A
the left translate of A to fu . The vector field A * on the principal bundle E � is said to be the fundamental vector field associated to A .
by A
P
generated
456
F I B E R B U N D L E S , G A U S S - B O N N E T , A N D T O P O L O G I C A L Q U A N T I ZA TI O N
17.2. Coset Spaces What do subgroups and easets have to do with fiber bundles?
17.2a. Cosets Let G be a Lie group and H e G a subgroup. The left coset space is the set f 0 equivalence classes of elements of G
g
g'
�
g'
iff
=
gh
for some h E H. Thus g � g' iff g- l g' E H. (We discussed this in the case of abelian groups in Section 1 3 .2c; we called it there the quotient space. In abel ian groups one may write g' � g iff g' = g + h .) gH = g'H
H
g '= g h g
G e I
I
1f
I I
I I
I
C/H
I
H
gH
Figure 1 7.2
Thus we identify all elements of G that lie on the same left translate
gH
:=
{gh lh
E
H}
of the subgroup H . We denote the equivalence class of the element g E G by [g] or else g H . The map that sends g into its equivalence class will be denoted by rr . Many familiar spaces are i n fact coset spaces ! Let u s say that a group G acts (as a transformation group) on a space M provided there is a map G x M � M
(g , x)
f--+
gx
such that
(gg')x
=
g (g'x ) and ex
=
x
If, furthermore, given any pair x , y , of points of M , there is at least one g takes x to y , gx = y , we say that G acts transitively on M . Example: S 0 (3) acts transitively on the 2-sphere
S O (3 ) as the group of rotations.
x
S2
�
S2
E
G that
457
COSET SPACES
Fundamental Principle (17.10): Let G act transitively on a set M. Let Xo and let H C G be the subgroup leaving xo fixed, H H is
=
{g
E
G lgxo
=
E
M
xo }
called the stability, o r isotropy, o r little subgroup of Xo. Then the points of in J : 1 correspondence with the left cosets {g H } of G.
M are
The s pace of left co sets is again written G 1 H . Unlike the case when G is abelian,
Gf H is usually not itself a group.
gH= g'H
H g
G
gH
e I I I I
I I I I
:rr
M
x = gxo
Xo
Figure 1 7.3 P RO O F :
Let Xo be a point of M. Associate to g E G the point x = gxo where takes the distinguished point Xo . Since ghxo = x also, for all h E H , we see that under this assignment, the whole coset g H is associated to this same x. We then have a correspondence GI H -+ M . Conversely, to each x E M we may associate { g E G : gxo = x } , which is easily seen to be an entire coset of G . D
g
Example: S 0 (3) acts transitively on the 2-sphere M = S2 . Let Xo be the north pole, Xo = (0, 0, 1 ) T . The little group of Xo is clearly the I -parameter subgroup of rotations about the z axi s . H for all e .
=
S O (2)
c
S O (3)
S O (2) =
[�
cos e si e
- sin e cos e °
O� l
We conclude that S O (3 ) / S O (2)
�
S2
In our usual picture of S O (3) as the ball with identifications, Figure 1 7 .4, S 0 (2) is the cu rve C. Note that all rotations through :rr about axes in the xy plane send Xo to the
sou th
pole. Thus the coset of the rotation diag ( I , - 1 , - 1 ) is the curve C'.
458
FIBER BUNDLES, GAUSS-BONNET, A ND TOPOLOG ICAL QUANTIZA nON
SO(3 )
_______ S2
Figure 1 7.4
The coset C ' is not a subgroup; it does not contain the identity. Note that in any (left) coset decomposition :rr : G -+ G / H, the subgroup H acts on G from the right as a group of transformations of G that sends each coset into itself
h
E
H sends g
E
G into gh ;
gH
f-+
g Hh
=
gH
The following i s a very important fact. We shall not prove this theorem here but we will make some comments about it.
Theorem (17.1 1): Let G be a Lie group and let H be a closed subgroup (i. e., H contains its accumulation points). Then G / H can be made into a manifold of dimension dim G - dim H. Furthermore, G is a principal bundle with structure group H and base space M = G / H and :rr : G -+ G / H is the projection of the bundle space onto the base. A coset space M = G / H of a Lie group is called a homogeneous space. For example, S 2 is a homogeneous space, being the coset space S O (3) / SO (2) of dimension 3 1 . Remarks on the Proof of Theorem (17.11): We indicate briefly why the cosets in a neighborhood of the coset e H = H can be considered a manifold of dimension dim G - dim H . -
Figure 1 7.5
459
COSET SPACES
Let V be an embedded submanifold of G, passing through e, transverse to H , and of di mensi on complementary to H (a "normal disc"). An essential fact that can be proved is that if V is sufficiently small, each coset g H of H will either miss V or else strike V in exactly one point. For this, it is important that the subgroup H be closed in G ; if, e.g . , H were a line winding densely on the torus G = T 2 of Section 1 5 .4d, then surely if H met the transversal V once it would meet it an infinite number of times ! ) . If a cose t o f H meets V w e may say that this coset is near H. A coset near H is o f the form g H for some unique g E V . This shows that the points of G / H "near e H" are in 1 : correspondence with the points of the "slice" V . Locally G / H is a manifold of the same dimension as V , that is, of dimension (dim G -dim H). For details see, for example, Warner's book [Wa] .
1
17.2b. Grassmann Manifolds The real projective plane lRp 2 is the set of unoriented lines through the origin of lR3 (S 2 is the set of oriented lines). The orthogonal group 0 (3) acts transitively on the space of lines. Let 1'0 be the x axis line. The subgroup that sends this line into itself consists of orthogonal matrices that either leave the x axis pointwise fixed or else reverses the x axis. These orthogonal matrices automatically send the yz plane into itself, that is, they act as does on the y z plane. Since 0 ( I ) consists of the two numbers { - I , I } , we can write the isotropy subgroup of 1'0 as
0 (2)
[0(01 ) 0(2)0 ]
= 0( 1 ) x
Thus lRP 2 may be identified with the coset space
0(2)
c
0(3)
0(2)
0 ( 3) __ 0(1) x The dimension of a cartesian product Mr x V S of manifolds is (r + s) . Thus lRP 2 is a manifold of dimension 3 - + 1 ) . The set of unoriented k-planes through the origin of lRn is called a Grassmann manifold and is frequently denoted by Gr(k , n ) . (Beware: there are different notations.) Thus Gr( l , 3) = lRP 2 .
lR P 2
=
(0
__
__
P roblems
1 7.2(1 ) Exh ibit G r(k, n) as a coset space and compute its d i mension.
Let 1'0 be the unoriented z axis. Show that we can as the coset space 8 0(3)/ H, where H is the subgroup C U C' consisting of the two curves C and C' considered in Figure 1 7.4.
1 7.2(2) 8 0(3) acts transitively on write � p 2
� p2 .
1 7 .2(3) We know that the collection of all frames of n orthonormal vectors at the origin
of � n can be identified with the group O(n) . Show more generally that the space of a/l orthonormal k-frames (f 1 ' . . . , fk) at the origin of �n forms a homogeneous space that can be written O(n) /O(n- k) . This space is called a Stiefel manifold. What does this say about 8n- 1 ?
460
F I B E R B U N D L E S , G A U S S - B O N N E T , A N D T O P O L O G I C A L Q U A N T I Z A T I ON
17.3. Chern's Proof of the Gauss-Bonnet-Poincare Th eorem
What is an "Index Theorem" ?
17.3a. A Connection in the Frame Bundle of a Surface
Let M 2 be an oriented surface with a Riemannian metric. Its tangent bun dle is an 12 bundle with structure group S O (2) . Although we could proceed with this real 2 -plan bundle, for our purposes it is more convenient to use instead the complex li ne bundl version of Section 1 6.3c. We shall, however omit the superscript c when dis cus sing the connection and the curvature. There should be no confusion since w an d e w ill carry no matrix indices, being 1 x 1 matrices of forms. Let then E = T M be the complex tangent line bundle to M 2 . As in Sec tion 1 6.3c , the structure group of this bundle is the unitary group V 0 ) , that is, the comp lex n umbers e ia of absolute value 1 . A frame at a point p is simply a unit tangent vecto r e at this point. Let FM be the frame bundle, with fiber and group the circle G = V ( 1 ) . (Note that in this simple case, FM is simply the unit tangent bundle to M 2 ! ) For g E G
�
( 1 7. 12)
Let eu be a frame, that is, a unit vector field on V (i.e., e u is a section of FM over V ) . As in Section 1 6.3c (and omitting the tensor product sign and the superscript C) , the connection form w is a single pure imaginary I -form
Ve u = e u ® w u = e u w u The fact that w is pure imaginary, that is, skew hermitian, arose because we demanded that parallel translation preserves lengths. To see this, consider the section eu . Let x = x (t) be a curve in V starting at x (O) . Parallel translate e u (x (O» along x (t) yielding a unit vector field (frame) e(t ) . Then e(t) = e u (x (t) g Ct) for some g et) E V e l ) ; that is, g et) is in the structure group. But then
Ve ( ve ) dg g+e = o = dt dt dt
( )
dX dg = ew - g + edt dt
and so
( 1 7. 1 3)
for this particular g = g et ) defining parallel translation along x = x (t ) . Thus the value of w on the tangent vector is - (dg/dt)g- I . But if g et) = e ia ( t ) , then - (dg/dt)g - I = -i (da/dt) is pure imaginary, that is, in the Lie algebra to V ( 1 )
w
( �; )
E �/ = � 1 )
( 1 7. 14)
v�c�or
It should not surprise us that (dg /dt)g - I is in 3' = � 1 ) since dg/dt is a tangent to G at g and g - I right translates it back into the Lie algebra. Th e Riemannian condition
461
C H E R N' S P R O O F O F T H E G A U S S - B O N N E T - P O I N C A R E T H E O R E M
r ((: 1
bundle connection demands that the connection Ijorm W takes its values in of the structure group. 1 3 we defined the general notion of a connection for a vector bundle E . 6. 1 n S tio In ec allowed u s to differentiate a section of E with respect to a tangent vector c onnection ase space M, that is, W is a matrix of local I -forms on M . Now we shall define 10 th e b a 1 x 1 m atrix w * of global I -forms on the 3-dimensional principal bundle space FM ! Let eu be a section of FM over U , that is, a frame on U , and let f be another section.
u 011 O
he Lie algebra
The
Then
f(x ) = eu (x ) gu (X )
E U ( 1 ) . The local "coordinates" for f are then (x , a), where a is the iable in ( 1 7 . 1 2) for g = gu o The local coordinates for e u are (x , g = e), var angular
for som e gu (x ) i.e., a
=
O.
____ (
u
)
__
____
M
Figure 1 7.6
Then Vf = V (eu g u )
=
eu w u gu + eu dgu
=
l e U gug{j w Ugu
Vf = f 0 {g u 1 wu gu + l
But gC; wu gu
an d so
=
g u 1 dg u }
+
l eu gu g{j dgu , or
Wu (it is crucial here that U ( 1 ) is commutative) and Vf = f 0 { wu +
ida }
( 1 7 . 1 5)
g- I dg
=
ida,
( 1 7 . 1 6)
Note that da can be considered a local I -form on the frame bundle since a is a local coordinate in JT - 1 U. JT * W is also, but we usually simply write w for JT * w since X l , x 2 , a are local coordinates for JT - I U i
w = r j dx = JT * w
for some functions r i on U . Thus we can define the local I on rr- 1 U by w * u := Wu +
ida
x
I matrix of I -forms w *
( 1 7 . 1 7)
Since w * is again pure imaginary, this is now a I x I matrix of I -forms on JT - 1 U C F M that still takes its values in u{ 1 ) . Now notice something remarkable. Since V f has a geometric meaning independent of the frame used (in fact, using the real forms, putting w + i da = - i w l 2 + ida =
462
F I B E R B U N D L E S , G A U SS-B O N N E T , A N D T O P O LO G I C A L Q U A N TI ZATIO
-i {W 1 2
. .
- da }
surpnsmg.
=
N
0 defines parallel translation see (9,62)), the follow ing sho uld n
��
Theorem (17.18): On an overlap wu * = wv*, and thus the collec tion { eu *} defines a W = Lt( 1) valued i -form w* on all of the principal bundle FM. eu� is called the connection form on the frame bundle FM. Let ev be a section over V . e v = eu c u v where C u v = e ifJ , for s ome fJ · Then a section f has two representations f = eugu = e v g v , where g u == eia and gv = e i (OI - fJ ) . Then at the point f of F M w* v = W v + id ea - fJ) . But Wv = cu � wu c u v + c u � dc u v = Wu + e - ifJ de ifJ = Wu + i dfJ . Thu s
PROOF:
w* v
=
Wu
+ i da
=
w*u
Thus, although {wu } are only locally defined �/ -valued I -forms, and { dau } are only locally defined I -forms, the combinations {w� } match up to defi ne a global 3rvalued I -form w * on FM . D But then
8*
:=
d w*
n *dw
n*8
( 17. 19) is also globally defined on FM; we shall call this the curvature form on the frame bundle. It is not new to us that 8 * is globally defined on FM since we already knew that 8 = i81 2 = i K (Y I ;\ (Y2 is globally defined on M2. What is new and so i mportant =
=
is Chern's observation;
Theorem (17.20): The lift n * 8 of the curvature 2-form to F M2 is globally exact on FM 8*
=
n *8 1 2
=
n * i K (Y I ;\ (Y 2
=
dw*
We have seen that IM 812 usually does not vanish, and thus 8 itself on M is usually not exact !
17.3b. The Gauss-Bonnet-Poincare Theorem
Theorem (17.21): Let M 2 be a closed Riemannian suiface and let v be a vector field on M having a finite number of singularities at P I , . . . , P N. Then I
-
2n
JiM
K a 1 ;\ (Y 2 = x ( M 2 ) = "" L ]v. ( POl ) 01
Note that since the left-hand side is independent of v, so is the right-hand s ide. This is Poincare's theorem ( 1 6.9). Since the right-hand side is independent of the abstract Riemannian metric used on M 2 ,
11 K d
A
must be independent of the metric.
C H ERN'S P R O O F O F T H E G A USS-BONN ET-PO I N C A R E T H E O R E M
Thi s is the
Gauss-Bonnet theorem. In Problem (8.20) we proved this for an 2 JR. ' . bedded s urface M C
463
em
The p ro of we shall give is due to S . S . Chern, who proved a far more general result. h ll talk about some of these generalizations later on. Chern's proof shows the e W s a equ ality of the integral with the index sum; we have already shown that the index sum l is the Eu er characteristic in ( 1 6.9). PRO
OF :
We shall prove the theorem when M is orient able (and oriented) ; the nonorientable case can be handled by the standard trick of passing to the 2-sheeted orien table covering, discussed in Section 1 6.2b. First remove small discs { Da } centered at the singularities. Then f = vI l vl is U D" . We then have a section a unit vector field, that is, a frame on M -
f : M2
-
U Da
--+
Cl
Figu re 1 7.7
F M2
464
FIBER B U N D L E S , GAU SS-BONNET, A N D TOPOLO G I C A L
Q UA N T I ZA TI
ON
(Remark: The frame bundle on this M 2 is clearly the same as the u n it tan . g bun dl e; fior h19· her-dImenSlOna ' . If. IS. Important I genera I IzatlOns · · to keep the fi ent
�
w� � �)
Let
be the image of the punctured M under the section f; it is a 2-dimensional sub manifold of FM diffeomorphic to M - U Da , since n o f is the identity map. Then since w* (not w) is globally defined •
i J'rJM-UD" KdA = i J'r KdA = Jr r� n*(iKdA) /rr = JrJ� :w* = la� ! = la� n*w - da
( 1 7.22)
Let the disc Da lie in the coordinate patch V and let Sa = a Da . Let eu be a frame in the open U . We may express the Wu of this frame in terms of the local coordinates x I , x 2 (which are unrelated to the frame eu), The part of the boundary of 2: that lies over D o is over Sa ; call this portion of a 2: simply aa ' Then in ( 1 7 .22) 1 n * w = 1 w = 1 Yi (x)dx i ha(l Is" La and if we let the disc Da shrink down to the point Pa this last integral will vanish in the limit. Thus as each Da shrinks to its Pa
JrJrM i KdA
=
lim
JrJrM-U D" iKdA = - lim fa� 1 ida
( 1 7.23)
Consider again the part of a 2: that lies over Sa . In terms of the section eu given by the frame, the section f is f = e u ei a . Note that the part of a 2:: that lies over Sa has orientation opposite to a Da (whose normal points out of Da ) , a 2: = 2: a f( - Da ) . Furthermore aa
1 d a 1 d L ( e u , f) !rcaD,,) faDa is simply 2n (index of v at Pa l = 2nj(Pa)' Then from (17.23)
( 1 7.24)
=
JrJr KdA = - lim 2: I-fCoDa 1 . de = 2n 2: jv(Pa) ) a
a
Corollary (17.25): If M 2 is a closed Riemannian manifold then
_1_ Jr r
2n JM
K dA is an integer
0
465
LIN E B UNDLES, TOPOL O G I C A L QUANTIZATION, A N D BERRY P H A S E
17.3c. Gauss-Bonnet as an Index Theorem frOm Probl em
1 6.2( 1 ) we know that the Euler characteristic
X
(M2 ) =
2 - 2g is
pre ssible in terms of the genus g of the surface. In Section 1 3 .4 we showed that
eXclosed orientable surface of genus 2 has first Betti number b l = 4, and we have �ndicated the generators A , B , C , and D. The same type of picture shows that a closed of genus g has b l ( Mg ) = 2g . If we recall that bo = 1 (since Mg is �entable surface that b2 = 1 (since Mg is closed and orientable), we see that the Euler and d), connecte (defined in ( 1 6. 1 1 » can be written a char cteristic ( 1 7.26)
in terms of homology ! This, and its n-dimensional version, was proved by Poincare. We shaH di scu ss this further in Problem
theorem (17.2 1 ) in the form
22.3(2). Finally we may write the Gauss-Bonnet
Jrr
1 _ KdA = bo - b l + b2 2 7r JM
( 1 7.27)
left-hand side we have a curvature, a local quantity involving derivatives of the On sor, quantities associated to the tangent bundle of M . Its integral (divided by ten tric me 21r) simply a number. The right-hand side exhibits this number as an integer involving
the is
of homology groups of M. Recall from Hodge's theorem that b p (M) is also equal to the dimension of the space of harmonic p-forms, which is nothing other than of the kernel of the Laplace operator
dim ensions
the dimension
bp ( M )
= dim ker �
:
p
/\ ( M )
-7-
p
/\ (M)
In ph ysics, the kernel of an operator is called the space of zero modes. Thus, basically, an integral of the curvature of the tangent bundle of M is related to the number of
zero modes of differential operators constructed from this bundle. This is the first and ost famous example of an index theorem. The Atiyah-Singer index theorem is a m
vast generalization of ( 1 7 .27) replacing the tangent bundle by other bundles (we shall consider a few examples in our next section), the Gauss curvature by higher-dimensional curvature forms (some of which will be discussed in Chapter 22), and replacing the Laplacian by other elliptic differential operators associated with the bundle in question. The Atiyah-Singer theorem must be considered a high point of geometrical analysis of th e twentieth century, but i s far too complicated to be considered in this book. The reader consult for instance, [Ro] .
may
17.4. Line Bundles, Topological Quantization, and Berry Phase
How does a wave function change under an adiabatic transition?
17.4a. A Generalization of Gauss-Bonnet
Let E be any complex line bundle over a manifold M" of any dimension. We suppose that the structure group G is U ( l ) ; 1/rv = eict 1/r u . Let w be a U ( l ) connection; that is,
466
F I B E R B U N D L E S , G A U S S - B O N N E T , A N D T O P O L O G I C A L Q U A N T I ZATIO
N
w takes its values in 3/ = LL( I ) , Thus w (X) is skew hermitian (pure im agin ary) f or all tangent vectors X to M il , If w = e u 1/Iu and cp = e u ¢ u are sections, then, allows us to define a hermitian scalar product in each fiber by ( W , cp) : =
1/I u ¢ u
with associated norm I W 1 1 2 = 1 1/I u 1 2 , We then say that E is a hermitian line bund!e, If we put, as usual, V 1/1 = d 1/1 + w1/l, then ( V W , cp) + ( w , V cp ) = d1/l¢ + 1/Id¢
+ w1/l¢ + 1/Iw¢
=
d(1/I¢)
since w = -w. This is the analogue of the basic Riemannian condition that Parallel translation preserve scalar products. Let eu be a "frame" over U , that is, a section of E of norm 1 . Then W = eu 1fru == e u e l a is the most general frame over U . Thus over U , the fiber coordinate in the frame bundle FE, that is, the principal bundle associated to E, is simply the angle a . The frame bundle is a circle bundle over Mn , a bundle whose fibers are circles S I . We may now proceed as we did in the case of the tangent bundle to the surface. Let V2 be a closed oriented suiface embedded in Mn . The part of the bundle FE over points of V 2 defines a bundle over V2, which we shall again call FE; it is the same circle bundle but "restricted" to V 2 • We wish to consider a smooth section W of FE over the closed surface V 2 , but we know from the tangent bundle case that such a section might not exist over all of V2 • We might try to construct such a section by first taking a section s : V 2 --* E of the complex line bundle, and then putting W = sf I s II at those p E V 2 where s =f. O. A section s defines a 2-dimensional submanifold S ( V2) of the 4-dimensional manifold E v , the part of the bundle E over V . The O-section 0 defines another 2-dimensional submanifold o( V2 ) of E v . Generically, a submanifold V' and a submanifold WS in an N -manifold, if they intersect, will intersect in a submanifold of dimension (r + s - N ) , just as in the case of affine linear subspaces of a vector space. The section s and the section 0 are generically then going to intersect in a O-dimensional set, that is, a finite set of points, which may be empty. Thus, just as in the case of the tangent bundle, we expect to be able to find a nonvanishing section of E, and a res ulting
section of FE, over all of V 2 except perhaps over a finite set of points
PI , . . . ,
PH '
(The precise argument for such constructions will be taken up in Chapter 22.) Let then W be such a section. As in Section 1 7 .3b we construct the connection fonn w* = rr * w + i dct, where ct is the local fiber circle coordinate (recall that w is now pure imaginary). Then, as in ( 1 7 .24), we define the index of W = e1/l = ee i a at the zero Pk to be j\jl (pd
:=
_ 1 J dct 2rr !a D
LINE BUNDLES, TOPOLO G I C A L QUANTIZATION, A N D BERRY PHASE
bicb i s si mply the degree of the map 1/1 : a Dk ---+ eorem ( 1 7 . 2 1 ), we conclude that (i /2;r ) ffv e 2 = sketched a proof of the following theorem of Chern .
S I . Then, just as in the proof of L jq, ( p) is an integer! We have
�
e Theor m ne ction
467
(17.28) : Let E be a hermitian line bundle, with (pure imaginary) con
Wi and curvature e 2 , over a manifold Mil . Let V 2 be any closed oriented
sur/ace embedded in Mil . Then
� /1 e 2
2
is an in teger and represents the sum of the indices of any section s : V 2 ---+ E of the part of the line bundle over V 2 ; it is assumed that s has but afinite number of
ze ros
on V . i e /2;r is the Chern form of E.
Thi s then proves Dirac 's quantization condition ( 1 6.52).
Geometrically this integer represents (algebraically) the number of times that the section s intersects the O-section 0, counted with multiplicity. By this we mean the
following: Let E be a rank-n vector bundle over an Mil . We assume that M is oriented and that the vector space fibers of E can be oriented in a continuous fashion. (This will be the case if the structure group G of the bundle is a connected group, such as S O (n), or a unitary group. On the other hand, as discussed in Section 1 7 . 1 a, the real line bundle given by the normal vectors to the midcircle M 1 of the Mobius band has structure group given by the 2-element group Z2 , which is not connected, yielding l fibers that cannot be oriented continuously.) Let X , , X " be positively oriented local 1 coordinates in M , and u , , u n be positively oriented fiber coordinates, near the intersection point x = 0, U = 0, of the sections s and o. s can be described by the n functions u = u (x ) . We say that the section s meets the 0 section transversally (or that s has a n o n degenerate zero) if the Jacobian determinant a (u ) / a (x) is nonzero at j x = O. From du j /dt = (au /ax k ) (dx k /dt) we see that transversality simply means that the sections do not have any nontrivial tangent vector in common at the intersection. In this case we define the local intersection number to be + 1 (resp. - 1 ) provided the Jacobian is positive (resp. negative) . The (total) intersection number is the sum of all the local intersection numbers at all intersections of the sections. Consider, for example, a complex line bundle E over the Riemann sphere. We may use z = x + iy for local coordinates on V 2 = S 2 near z = ° (S 2 is a complex manifold) and I; = u + i v for fiber coordinates. The section s can be described by giving u = u (x , y), v = v ex , y), or more briefly I; = I; (z, z) , where we do not assume that I; is holomorphic in z. If, however, I; is a holomorphic function of z (we would then say that s is a holomorphic section) then by the Cauchy-Riemann equations we have a (u , v)/a (x , y) = I I;' (z) 1 2 :::: 0. Thus if a holomorphic section is not tangent to the O-section, t;' (0) i- 0, we conclude the local intersection number is + 1 . Consider as a specific example the tangent bundle E = T S 2 of the Riemann sphere as a complex line bundle. Use z as coordinate near ° on S 2 and w as coordinate near 00 . Let I; be a fiber coordinate over the z patch. On the Riemann sphere we have the vector field coming from dz/ dt = Z 2 . It has (Kronecker) index j = 2 at z = ° and ° at z = 00 .
•
•
•
.
•
468
F I B E R B U N D L E S , G A U S S -B O N N ET , A N D T O P O LO G I C A L Q U AN T I ZA TI
ON
Wse i:titoons? ectio : eld seen� �� coale:e
(since dw/dt = - 1 at z = (0 ) . How can we think of this in terms of inters . The part of T S 2 over the z patch is simply ((: 2 with coordinates (z , O . We � how the section ( = ( z) = Z 2 intersects the section ( = O. Clearly these tangent (i.e., nontransversal). B y a slight deformation, however, we m ay r pl ns � section by one with transverse intersections; for example consider the secti on this by ( = Z 2 - a (i.e . , dz/dt = Z 2 - a ) , for some small a # O. Near Z =: oo the fined is dw /dt = - 1 + a/z 2 , and again has no zero; the zero at Z = 0 has been repl by two zeros at the square roots of a . In this holomorphic case, as we have zeros have local intersection number + 1 , yielding +2 as the total intersection n m of the perturbed section with the O-section. As we let a -+ 0 the two zero s and in this sense we say that the original section meets the O-section with intersec · on number 2 . This agrees with the Kronecker index j . Note that this is very different from the usual real situation. In the real plane 2 R the curve y = x 2 is tangent to the x axis, but if we lift the curve slightly y == X l + a (for a > 0) there is then no intersection at all. On the other hand, if We drop the curve, y = x 2 - a , we get two intersections but the intersection at x =: ..;a is I whereas that at x = - Ja is again yielding a total intersection number O. For a good discussion of Kronecker indices and intersection numbers I reco mmend [G, p. chap. 3 ] . Return now t o the general situation o f Theorem ( 1 7. 28). Note that when the Betti number b2 of Mn is zero, for example, when each closed surface V2 bo unds, the integral condition in ( 1 7 .28) is simply ffv =iJ S e 2 = fffs de 2 = O. The integer in ( 1 7. 28 ) can be nonzero only when Mil has nontrivial homology in dimension 2; ( 1 7.28) is a
to
-I,
second
topological quantization condition.
We may paraphrase ( 1 7. 28 ) as follows. The curvature e of a hermitian complex line bundle is a pure imaginary closed 2-form on the base space Mil having the property that i B /2JT has integral periods on any basis of the integral second homology group H2 (Mil ; Z) . We say that ie /2JT defines an integral cohomology class of M. There is a remarkable converse to this, whose proof is beyond the scope of this book.
Theorem (17.29): Let f3 2 be a real, closed 2-fonn defining an integral cohomol ogy class on some manifold Mn. Then there exists a hermitian line bundle E over M, and a U ( 1 ) connection w for this bundle, such that -2JT if3 is the curvature form on M for the bundle E. Thus each closed 2-form on M with integral periods is essentially the curvature form for some hermitian line bundle over M. Furthermore, one can define a notion of "equivaleDl bundles" and then if M is simply connected, the constructed line bundle E is unique (up to equivalence) ! The proof requires the introduction of more machinery (sheaf theory) and will not be given here. 1 7 . 4b.
Berry Phase
We are going to be concerned with complex line bundles, but a real example will give us a good picture to start out with.
LIN E B UNDLES, TOPOLOG I C A L Q U A N T I Z A TI ON, A N D B E R R Y P H A S E
469
E(O)
Figure 1 7.8
Consider an infinite Mobius band (i.e., each generating straight line is infinite) immersed in IR 3 , with central circle V I given by x 2 + y 2 = 1 , z = 0 and parameterized by () . The infinite real line of the Mobius band passing through (cos e , sin e , 0) can be identified with a real I -dimensional subspace of IR3 by translating it to the origin, yielding a I -parameter family of real I -dimensional subspaces Eo of IR3 . We can pick out smoothly a real unit vector e(e) in Eo in some neighborhood of e = 0 (unique up to multiplication by ± I ), but since e(2n ) will be the negative of e(O), we see that we can't find e(e) smoothly for all e . Look now at the following more general geometric situation. Consider a complex inner product space; in our main example it will be infinite-dimensional but for easy visualization we take en with the usual hermitian scalar product (z, w) = I: Zj W j . Sup pose that for each point p in a K -dimensional parameter manifold V K , we may assign acomplex I -dimensional subspace ("line") E p of en . We thus have a K -parameter fam ily of complex lines. If a = (a I , . . . , a K ) is a local coordinate system for V , we may describe the family by Ep = Ea . We assume that the lines El' vary smoothly with p, an d so locally Ea depends smoothly on a . Each Ea is simply a copy of the complex plane C, and of course we can pick out a uni t basis vector e( a) in each Ea, and e( a) is unique up to multiplication by a complex number e i r (a ) of absolute value 1 . S ince Ea varies sm oothly with a , in some a -neighborhood of, say a = 0, we may pick the bases e(a) to vary smoothly with a . We may assume that the coordinate patches a are so small that ea is smooth in the entire patch. The family E p forms a complex line bundle E ove r the
K -dimensional parameter space.
local section of this bundle is simply a complex vector field v = e( a) u (a) . defi We ne a covariant differentiation by simply taking the projection of the usual derivative in
470
F I B E R B U N D L E S , G A U S S- B O N N E T , A N D T O P O L O G I C A L Q U A N T I Z A TI ON
In terms of the basis
Vv
e(e, dv)
=
Thus
=
( :)
e e, a k dC/ a
Ve(a) = e (a) (e(a) , de (a» ) = e (a)w l where
Wi
( ;)
(e (a) , de (a» ) = e, a k d a k a
:=
07 .30)
Note that this would not be useful in the case of a real line bundle since a e (rx ) /oaj is orthogonal to e (a) in that case. In our complex line bundle, however, 0 =
d (e, e) = (de, e) + (e, de)
=
2Re (e, de)
shows not that (e, de) vanishes, but only that it is pure imaginary. In a coordinate patch overlap we have e ( fJ ) = e ( a )cet/! for some functi on ca/! (p) of absolute value 1 , and so our bundle E is a hermitian bundle, with structure group U(1) and connection w. See Problem 1 7 .4(2) at this time. The curvature of this connection is given (see Problem 1 7 .4(3» by
e2
=
dw
=
d (e(a ) , de (a» )
meaning
e2
=
'
I
1m
=
(de (a) , de (a» )
)
/ a e (a) a e (a » da j 1\ d a k j k
( 1 7.31)
\ oa ' o a Then Theorem (1 7. 28) gives topological quantization conditions in this purely geomet ric situation !
In Section 1 7 .4c we shall apply this machinery to Dirac 's monopole bundle, but for the present we shall consider examples investigated by Berry in [B]. First we shall study a finite dimensional situation. = H (a) be an n x n hermitian matrix that depends smoothly on K -parameters a j . (We may think of this as perturbing a given hermitian matrix H (0) . ) H (a ) operates on <e" and has n real eigenvalues for each a . We shall assume that the lowest eigenvalue A I (a) lor H (a ) is nondegenerate for each a ; thus there is a unique complex I -dimensional eigenspace Eet C <en picked out for each a. We assume that the set of lowest eigenvalues { A I (a) } is separated from the higher eigenvalues. Note first that A I depends smoothly on a. To see this, observe that the characteristic polynomial I (A , a) = det[A ! - H (a ) ] is a smooth function of both A and a. Fix a = a D , and let A] be the unique lowest eigenvalue; thus I (A I , a n ) = O. Since A I is a simple root, we have Example : Let H
=
H (a I , . . . , a d
o
and A I differs from A j for j i= 1 . Hence 1 = 0 and afl a A i= 0 at A = A l and a == a . . From the implicit function theorem we conclude that A I is a smooth function of a In some a neighborhood of a D .
LIN E BUNDLES, TOPOLOGI C A L QUANTI ZATION, AND BERRY PHASE
471
It can be shown (see [Kal for more details) that the I -dimensional eigenspace E(){ of the l ow est eigenvalue A I (a) also depends smoothly on a. A sketch is as follows. S ince = L j Aj (a ) Pj (a ) , where Pj (a) is s H (a) i hermitian we may write it in the fo rm H (a ) gonal projection onto the eigenspace for A j (a) . Hence for any complex number rtho o the = [Aj (a) zI z l Pj (a ) . Then for the resolvent [ H (a ) - ZJ]- I L we ve H (a) ha j : we have Thus if C is a closed curve enclosing positively the set of lowest eigenvalues {A I (a ) but excluding the higher eigenvalues, we have, for each a
i
[ H (a ) -
zJ]- l dz
=
L J
=
{i
dZ/ [ Aj (a)
- 2n i PI (a)
-
Zl } Pj(a)
exhibiting P I (a) as a smooth function of a. Thus the first eigenspace, image of C" under PI (a ) , is smooth in a . D
E(){ , being the
Again a unit e i gen ve c tor e(a) for A ) (a) is determined only up to multiplication by number of absolute value I , as in our general situation. Berry considered the following infinite-dimensional quantum situation. Let :'If be a complex "Hilbert space" of functions on Mil with hermitian scalar product (
a complex
(
=
L 1jf (x )
S tates 1jf in quantum mechanics are normalized, (1jf 1 1jf ) = 1. The wave function for is determined up to multiplication by a constant factor ei!.. of absolute
a state classically
value 1 .
Berry considers a quantum analogue of our example in which a Hamiltonian operator = H (a ) , acting on :-1(, depends smoothly on the points a in a K -dimensional param eter space V K . Locally the point a is again described by coordinates a = (a i , . . . , a K ) . (For example, i n the Aharonov-Bohm situation a = a 1 could be the flux b through the solenoid, or we might have several solenoids with such varying fluxes. ) The spec trum of H (a) is assumed to satisfy the requirements of our example. We again are led to a complex line bundle E over space-time M 4 , whose fibers are the complex I-dimensional subspaces E(){ C :'Ie given by the eigenspaces of lowest energy of H (a) . Now let C b e a curve i n parameter space, locally of the form a = a Ct ) , starting at a = O. Consider a solution 1jf (x, t) of Schrodinger's equation i1id1jf/dt = H [a (t)] 1jf on M that starts out at t = 0 with 1jf (x, 0) a lowest energy eigenfunction of H (O) . The adiabatic theorem [Si] assures us that in the limit of a changing "infinitely slowly," the s olu tion 1jf (x, t) will remain an eigenfunction of lowest energy of H (a (t » . If the curve a = a C t ) is a closed curve, the solution 1jf will then return to 1jf (x, 0) except for a ph ase factor, and this phase factor was determined by Berry as follows. H
472
F I B E R B U N D L E S , G A U S S- B O N N E T , A N D T O P O L O G I C A L Q U A N T I Z A T ION
Let ¢a H ( a ) ¢a
C =
Ea C :Ie be a smooth choice of unit basis of Ea in the a patc h; ¢a satisfi Aa ¢a and replaces the e(a) of our previous example. Fro m the adi ab c:s
t�
theorem, for very slowly changing a C t) , o/ (t) can be approximated by a mU1tipl e o eigenstate ¢a(I ) . Berry writes, for this particular path C in parameter space ,
0/
"-'
exp
[ ( - �) 11
]
Aa ( r ) d r exp[ i y ( a (t» ] ¢a (l )
( 1 1 . 32 )
(For a more careful treatment of the adiabatic limit see [Sil .) The energy exponential is the usual dynamical phase factor (taking into account the fact that A i s c hangin in time along the path) and the second (as y �t unknow � ) exponential exp[i y (a Ct)}] . . to account for the bases ¢a havmg rather arbltranly assigned phases. Inserti ng ( 1 1 32) . into Schrodinger's equation yields Berry's equation
.
l ¢a
dy dt
or
+ (�) ( ) da j dt
a a l·
i�
=0
as :k-valued I -forms a lo ng C in parameter space, where d (¢ I d¢ )
=
-i
=
da j a / aa j . But then
dy
Barry Simon [Si] noticed that this can be written down in terms of connections. From ( 1 7 .30) we have, along C,
dy
=
iw
( 17.33)
where w is the connection in terms of the frame ¢. We shall call w the Simon connection (avoiding the temptation to call it the Berry-Barry connection). Thus if C is a closed curve in a coordinate patch of parameter space, and if C bounds, that is, C = a S for a compact oriented surface S in this patch, then the Berry phase factor for C is given, from Simon's viewpoint, as
y (C)
=
=
1 w i Jis e a -1m Jr r / � I ) da j is \ aaJ aa
1
dy =
i
( 1 7.34)
=
¢
k
/\
da k
Note in particular that y (a) need not return to itself after completing a loop in parameter space, and likewise, neglecting the dynamical phase factor, for the wave function 1/1. This was one of Berry 's principal conclusions, and it should be mentioned that the final expression in ( 1 7 .34) appears explicitly in B erry's paper but is not there related to curvature. In Problem 1 7.4(4) you are asked to show that e i Y ¢a is paraUel displa ced along C . This gives geometric meaning to Berry 's ansatz ( 1 7 .32) and also to the adiabatic theorem. For an application of the connection ( 1 7 .30) and the quantization condition (17. 28) to the "quantum Hall effect," see [Si] .
LINE BUNDLES, TOPOLOGICAL QUANTIZATION, A N D BERRY PHASE
473
17.4c. Monopoles and the Hopf Bundle
In Secti on 1 6.4 we discussed the Dirac monopole, which, for each integer n = 2eq /n , require s a special hermitian complex line bundle, Hn , defined over JR. 3 with the origin deleted. The unit sphere S 2 surrounds the monopole, and for our purposes it is sufficient w c onsider the part of the bundle that lies over S2 , which we shall again call HI! . The
case n :::: 0 c orresponds to the trivial bundle; the most important case is when n = ± 1 , that is , when q = ± 1 /2en . We shall look at the case n = - 1 . This complex line bUndle H- I over S 2 is not the tangent bundle since the integral of i e 2 / 2Jr over S2 is - 1 , whereas for the tangent bundle the integral is the Euler characteristic 2 . It is remarkable that Hei nz Hopf investigated the appropriate bundles for purely geometric reasons (as we shall see in Section 22.4c) at about the same time as Dirac's work on monopoles ! Consi der S 2 as being the Riemann sphere, that is the complex projective line ee p l of S e ction 1 .2d and Problem 1 .2(3). To a point (zo , Z I ) #- (0, 0) i n ee2 we associate the line ( A z o, AZ I ) of all complex multiples of this point. This line is described by the point in eep I whose homogeneous coordinates are [zo , Z I ] . In the patch U of S2 where Zo :f. 0 we introduce the complex coordinate Z = Z I / zo, whereas in V , where Z I #- 0, we use W = Z O /Z I . The complex lines through the origin of ee 2 are parameterized by the points of ee p i d an thus these lines form a complex line bundle over S 2 , called the Hopf bundle. In a sense thi s bundle is "tautologous"; a point in ee p I represents a complex line in ee 2 , and we may then associate to this point its complex line ! Let us look at the local product structure .
When Zo #- 0, the line through (zo , Z I ) has homogeneous coordinates [ z o , Z l ] = [ I , zd zo] = [ l , z ] . To the point in U C S 2 with coordinate z , we may associate the vector ( 1 , z) T in this line of ee 2 . We call the resulting unit vector
eu (z ) =
( 1 , zl
( 1 7.35)
( 1 + I z I 2 ) 1 /2
This defines a unit section of the part of the Hopf bundle over U in S2 . Likewise, over V we have [zo , z I J = [zo /z I J = [ w , 1 ] with section
ev ( w ) =
(1
(W, l )T I w I 2 ) 1 /2
+
( 1 7 .36)
Thus the transition functions e v = e u cu v are given through
e
V
(w)
w ( l , W- I ) T
(w, I)T ----:-------,- ( 1 + I w I 2 ) 1 /2
-
(1
z- I ( 1 , Z) T
+
I w 1 2 ) 1 /2
e u (z ) l z l
( 1 + I z l - 2 ) 1 /2
Z
Thus c v u (z ) _
Z - - e i ¢> Iz _
i
( 1 7 . 37)
z = I z l e i ¢> = re i ¢> in terms of polar coordinates in U , that is, the upper plane F ig ure 1 . 1 5 . These transition functions are exactly those of the monopole bundle in
Where
474
FI B E R B U N D L E S , G A U SS-B O N N E T , A N D T O P O L O G I C A L Q U A N TI Z AT I ON
1 as we see from Equation ( 1 6.5 1 ). The monopole bundle li-I lV/tit. 2eq In = 1 is the tautologous Hop! bundle over C p l . HI will have transition functions C v u = e- i q, . This is the dual bundle to 11 Consider now the tensor product bundle of H_ I with itself, H- I ® H- I · This te� with 2eq ln =
-
,
product of two line bundles is again a line bundle; if s and I] are sections of li_ 1 t"� ' '''1:11 (s l]) v = (cv u su ) (cv u l]u) = e 2'' q, (s l])u shows that H_ I ® H_ I = H_ 2 . In thi s WaY we can get all of the monopole bundles from tensor products of HI and H- I , th at i s, frotn the Hopf bundle over CP I and its dual. We may now consider the Simon connection for the H�pf bundle. C 2 carries the stan. dard hermitian metric ( a , b) T , (c, d) T ) = ac + bd. Let us compute Wu :::: ( e u ( z ) , de u ( z» ) in the patch U . Note that U is simply a copy of the complex Plane. Introduce polar coordinates z = re'q, . In Problem 1 7.4(5) you are asked to compute. from ( 1 7.35), that
( 1 7.38) eU = dcv U = and
2i rdr 1\ d
( 1 7.39)
1
07.40)
----
jr r i e = Js 2n
-
P roblems ------
1 7.4(1 )
1 7.4(2) 1 7.4(3) 1 7.4(4) 1 7.4(5)
Take as line bundle the tangent bundle to the Riemann sphere. Let 4>z (resp. tPw) be a fiber coordinate in the z ( resp. w) patch. Show that the transition function is Cwz = _ Z -2 . Since l 4> w l 2 # l 4>z l 2 , we do not get a herm itian metric in the fibers by defining II 4>z 1 1 2 = l 4>z l 2 , and so on. It is true that I w l -2 1 4>w I 2 = I z l -2 1¢z I2 but these "metrics" blow up at the poles. Show that ( 1 + I z I 2 ) -2 1 4>z I 2 = (1 + I Wl2r2 l 4>w I 2 . This expression then yields an Hermitian metric in the fibers. Verify that w in ( 1 7.30) does transform as a U(1 ) connection . Show ( 1 7 . 3 1 ) . Show that e iy(a ) 4>a is parallel displaced along
C.
Show ( 1 7.38), ( 1 7.39) , and ( 1 7.40) . The integral over 82 C P1 is the same as the i ntegral over the entire U plane since only the single point at i nfinity is missing. =
CHAPTER
18
Connections and Associated Bundles
e
IN this chapter we shall recast our previous machinery of connections, making more use of the fact that the connection and curvature forms take their values in the Lie algebra of the structure group. This will lead not only to a more systematic treatment
of some topics that were previously handled in a rather ad hoc fashion, but also, in our following chapters, to generalizations of the Gauss-Bonnet-Poincare theorem and to closer contact with the machinery used in physics.
1 8. 1 . Forms with Values in a L i e Algebra
What do we mean by g - J dg ?
18.1a. The Maurer-Cartan Form
If E is a vector bundle over M, then the connection form ()) = «()) R s) and the curvature fonn () = «(} R s) are locally defined matrices of J - and 2-forms, respectively. If the Lie group G is the structure group of the bundle, that is, if each transition matrix Cu v = (c u v R s) is a matrix in G, then, as in ( 1 7 . 1 4), we usually require that ()) and f) take their values in the Lie algebra �/ of G ; thus, e.g., «()) R s ( X» is a matrix in �, for each tangent vector X to M. For example, in a Riemannian M, by restricting the frames of the tangent bundle to be orthonormal, the Levi-Civita connection satisfies ())ij = - ())j i ; that is, ()) has its values in a-(n) , the Lie algebra to O (n) . If we think of ()) as being a fonn that takes its values in the fixed vector space �/, rather than as a matrix of I -forms, we shall have an equivalent picture that is in many ways more closely related to the tenninolo gy used in physics. Let Mn be a manifold and let G be a Lie group with Lie algebra � . We shall consider loca ly defined exterior forms
�
475
476
CONNECTIONS AND ASSOCIATED B UNDLES
invariant I -forms o n G forming, a t each
E
g
G , a basis dual t o {XR } . Then
Q : = ER 0 a
R
( 1 8. 1 )
defined by R Q (Yg ) = ER a (Yg )
takes a vector Y = X R Y R a t
g
=
ER y
R
G and left translates it back to the identi ty. Thi s i s the
E
Maurer-Cartan I-form on G .
Classically this would be written differently. O n any manifold, Cartan wrote d for the vector-valued I -form at E M that takes each vector Y at into itself. coordinates it is the mixed tensor {o j }
p
Then Cartan would write
dp
�
p
= - 0
8 8xi
d
.
. . 8 0 0 1'. x 1 8xi
d
x' = -
( 1 8.2)
Thus
dg
takes Y at
g
into Y, and
g- 1
left translates Y back to e. We should write
For a matrix group, each Er is simply a matrix of a certain type (e.g., skew symmetric for G = O ( n ) . By construction we have the following:
)
Theorem (18.3): In any matrix ant I -form entries.
group
G, Q =
g- l dg
For example, in S O (2) , for
is a matrix with left invari
()]
- sin cos ()
g-l dg [ COS () () ] [- ()()dd()() ()()dd()() ] g- 1 dg [ d() -d()O ] [ 1] d() d() 1 ghg- l h - 1 hdg g- l dg dg g, (hg)- l d(hg) dgg- 1
we have
=
- sin ()
or
=
°
sin cos
sin cos ()
-
_
0 1
- cos - sin
-
°
0
and is a rotation invariant I -form on the circle S O (2) . The usual "proof" that is a matrix of left invariant I -forms is as follow s: Let == be a given (fixed) group element. Then for variable L7, Qhg = = Q g , as claimed. = Similarly is a matrix of right invariant I -forms. See Problem 1 8. 1 ( 1) at this time.
477
FORMS WITH VALUES IN A LIE ALGEBRA
IS.lb. �rValued p-Forms on a Manifold
The most general p-form on U
is of the form
C
M with values in the Lie algebra of a Lie group G
R where each ¢ is an ordinary exterior p-form on RU . Thus if X is a p tuple of tangent vectors to M at a point of U, then ¢ (X) = ER ¢ (X) is in �/. (Note that R refers to the ER inv olved, not to the degree of ¢ .) Since the E's do not vary (lying in the fixed vector space � ), it is natural to define ( 1 8 .4) a
-val ued p + 1 form on M . (�M ultipl ication of such forms i s not s o clear a process because, for example, in the case of �= o (n ) , the product of two skew symmetric matrices is not necessarily skew symmetri c. Instead we shall define the (Lie) bracket of forms, and we shall see that thi s includes a desirable product. We define ( 1 8.5)
As an example, consider the Maurer-Cartan I -form on M Cartan equations ( 1 5 .23)
=
G . From the Maurer
while
[Q , Q] and so
[Es 0 (J s , E 0 (J T ] T = E 0 C f (J S /\ (J T R T
=
1
=
dQ + 2: [Q , Q ]
[Es , ET] 0 (J s
=
/\
(J T
0
( 1 8 .6)
which will again be called the Maurer-Ca rtan equation. Remark : ( 1 8 .5 ) defines the bracket by means of a basis but it is not difficult to give an intrinsic definition. If X , = X I , . . . , Xp+q are tangent vectors to Mn , we could have defined ( 1 8.5')
R 8f K [ER 0 ¢ (Xj ) , E s 0 1/1 s ( X IS ) ] R = 8fK [ER , E S ] ¢ (X1 ) 1/I S ( X IS: ) = [ER , E S ] ¢R =
We have some immediate consequences of our definitions.
[ 1/1 , ¢ ]
= =
[E s , ER ] 0 1/1 s - [ER , Es] 0 1/I
/\
s
¢R
/\
¢R
/\
s 1/I (X, )
478
CONNECTIONS AND ASSOCIATED B UNDLES
and so ( 1 8.7) Also that is,
d [¢ , 1/1]
= [d¢ , 1/1]
+ (- I ) P [¢ , d1/l]
( 1 8. 8)
Finally, we need to interpret the bracket in the case of a matrix group. For ex ample, the Maurer-Cartan I -form Q for the affine group of the line, G = A ( l ) , is
[r
� 1 [� � ] ® d: + [ � � ] ® d: = E 1 ® dxx + E2 ® dy-x =
-
In general, when {E R } are matrices, [¢ , 1/1]
= [ER ' Es ] ¢ R !\ 1/I s R s = (ER ES -REs ER ) ® ¢ s!\ 1/I = (ER ® ¢ ) !\ (Es ® 1/I ) - EsER ® ¢R !\ 1/I s
where in the first term of the last line we are simply mUltiplying the matrices but using the exterior product of the entries. (This is always what we did in the method of moving frames, e.g., when considering e dw + w !\ w.) For example, in A ( l )
=
[ dx
x
o
o
!\
x
0
x
o
Continuing with our computation
that is, ( 1 8.9) as matrices. For example, if p is odd
[¢ , ¢]
= ¢ !\ ¢
+ ¢ !\ ¢
= 2¢ !\ ¢
as matrices. Note that if either ¢ or 1/1 is of even degree, then [¢ , 1/1] , as a matrix, is the usual commutator, but using the wedge !\ as product. If both are odd, then [ ¢ , 1/1] is
anticommutator
[¢ , 1/1]
= {¢ , 1/1 }
:=
¢ !\ 1/1 + 1/1 !\ ¢
the
479
FORMS WITH VALUES IN A LIE ALGEBRA
Consi der, for example, a Riemannian manifold with locally defined connection forms (j). We m ay restrict ourselves to the use of orthonormal frames, in which case w takes value s in
1 e = dw + 2: [ w, w]
( 1 8 . 1 0)
and of course is M (n) -valued. Likewise, the second Bianchi identity
de + w 1\ e - e 1\ w = 0
again makes sense in the Lie algebra setting since it says
de
+
[ w , e]
=
de + w 1\ e - (_ I ) l o2 e 1\ w = 0
(1 8. 1 1)
18.lc. Connections in a Principal Bundle
In Section 1 7 . 3 , a crucial role was played b y the notion of a connection i n the principal
bundle of frames to a Riemannian surface. Now we develop this machinery for the case
of the principal bundle of frames of sections of an arbitrary vector bundle. Let E be a real or complex rank- K vector bundle over a manifold Mn , the structure group being a Lie group G . Thus the transition functions cu v (x ) are linear transfor
mations of ]R. K or C K into itself. To say that G is the structure group means, effectively, that there is in each trivializing patch U of M a distinguished collection of frames of sections (e.g., orthonormal), which we may call G frames, and such that any two such frames eu and e v in an overlap u n V are related by e v = e u g for g E G . What do we mean then by a G connection? Let { w u } be a connection for this vector bundle. If e u is a G frame of K sections
Veu = eu 0 w u = eu wu where w�fJ = wffJ (x)dx i is a matrix of i -forms on U . Let C be any curve in U and let f b e a G frame at the single point C (0) . Parallel displace f along C. To say that w is a G connection is t o demand that the parallel displaced f is a G frame along all of C
and this must be true for all curves
If we now write, along C ,
C . Let f be such a parallel displaced G frame. f(t) = e u g (t)
we have, a s in ( 1 7 . 1 5),
[ -I ( )
]
Y'f dX _ dg ( 1 8 . 1 2) - = f0 g w - g + g Idt dt dt Since the entire frame is parallel translated along a curve x = x ( t ) in U , we must have dX _ _ dg g Iw g = -g I ( 1 8. 1 3) dt dt
( )
480
CONNECTIONS AND ASSOCIATED BUNDLES
where the term o n the right i s a n element of � . Then the first term
g- l W
(��) g
( 1 8. 14)
is also in � , and since Lg - 1 0 Rg certainly sends � into itself we have * *
w
(��) E �
Thus to demand that w is a G connection is to require that Wu is a �/ -valued 1 101"1rl on U . O f course the curvature i s then also � -valued.
eu
=
1
dwu + 2: [wu , wu ]
Under a change of frame
ev = eu c u v
( 1 8 . 1 5)
W v = cu- vI wu c u v + cu- vI d C u v ( 1 8. 1 6)
The transformation rule for e was exhibited in (9 . 4 1 ) for the tangent bundle; the proof here is the same. Consider now the principal bundle P of frames of sections of the vector bundle E . This fiber bundle has for fiber the structure group G and the transition functions Cuv are the same as for E; now, however, they operate on G by left translation,
F
g E G is sent to c u v (x)g We now define the connection forms w* in P ; these are � -valued I -forms on P rather than M . The local frame eu of sections of E can be thought of as a section of the bundle P over U . For a point f E P over the point x E U we can write
f = eu (x)gu (x)
( 1 8 . 1 7)
for a unique gu E G . From ( 1 8 . 1 2) we are encouraged to define
w � (x , gu )
:=
g i / n * wu (x)gu
+
g i / dgu
( 1 8 . 1 8)
which is the nonabelian version of ( 1 7 . 1 7 ) . We usually omit the n * coming from n : P ---+ M . The local section e u of P over U gives u s a local product structure U x G for in fact ( 1 8 . 1 7) assigns to the point f in P the local "coordinates" x in U and gu in G. A tangent vector at (x , g u ) in P is a velocity vector (dx j d t, dgu jd t) to some curve in P . grj' n *wugu applied to this velocity vector yields gr / wu (dxjdt ) gu , an element of � . grj' dgu applied to this same velocity vector yields gij' dgu (dgu jdt) = gu l dg U j dt , which is again an element of 3/. Thus wi; is a local 3rvalued I -form on P . Both termS in wi; depend on the choice of section eu .
n- 1 U;
481
ASSOCIATED BUNDLES AND CONNECTIONS
Theore m (18.19): In n - I (U n V) we have wlj = w� and thus { wlj } defines a global lJ' -valued I -form w * on the principal bundle P . (In P we may then consider the distribution o f n -planes transversal t o the fibers, defined by w* = O. This distribution is called the horizontal distribution, reminiscent of that appearing in the tangent bundle discussed in Section 9 . 7 . Many books take this
distribution as the starting point for their discussion of connections.) PROOF :
See Problem 1 8. 1 (2). We then define the global lJ' - valued curvature 2 - form e * on P by
e*
:
1
dw * + - [ w * , w * ] 2 = dw * + w * /\ w * =
( 1 8.20)
Note that unlike in the case of a tangent bundle of a surface (where the group G was abelian) we cannot expect e * to be globally exact or even closed! Of course we also have local curvature forms eu = dw u + Wu /\ Wu = dw + ( 1 /2) [ w , w] on M from the vector bundle connection. As in (9.39) one can show ( 1 8.2 1 )
P roblems
18.1 (1 ) Exh ibit the left i nvariant and the right invariant 1 -forms on the affine group of the line ( Example (i), Section 1 5 . 1 ) by means of g - 1 dg and dgg - 1 .
E ][-1 ( U n V) , f eu gu = e v g v , e v e ucu v . and so on. Use the transformation rule ( 1 8 . 1 6) for the vector bundle.)
1 8.1 (2) Prove ( 1 8 . 1 9) . (At a given f
=
=
18.2. Associated Bundles and Connections
What does it mean to take the covariant derivative of ,Jg?
18.2a. Associated Bundles Let P be a principal bundle over Mil with fiber = group = G, and with local transition matrices Cu v : u n V � G . Let p : G --+ G l ( N ) be some re pre se ntation of the structural group G ; thus each p (g) is an N x N matrix operating on iff and p is a homomorphism p(
gg ' ) = p (g)p (g') and p (g - I ) = [ p (g) r l
482
CONNECTIONS AND ASSOCIATED BUNDLES
(For example, w e may represent G = U ( 1 ) a s a subgroup o f GI(2) b y putti ng P (ei9) :::: diag(e1e , eM ) .) We then define a new vector bundle :rr : Pp � Mil associated to P th rou gh the rep resentation p , with fiber eN , by making identifications in U x eN and V x eN
(x , 1/rv )
�
(x , 1/ru ) iff
1/rv
=
p (cvu (x» 1/ru
Thus we construct a new vector bundle by simply using the new transition matrices p (c u v ) rather than cu v . We frequently have the following situation. Let E be a vector bundle :rr : E � AI with transition functions
cu v
:
V
un
�
each C u v (x) being a K x K matrix. Thus in and y v K-tuples of real or complex numbers)
G :rr
c
GI ( K )
-I (U
n V ) w e are identifying (for Yu
(x , y v ) � (x , yu ) i ff y v = c v u (x) yu We may then form the principal frame bundle P over M by considering the K- tuples of local independent sections e� (p) : =
ar
=
[ BxBxuv ]
T
ij
y a
=
[ Bxy ]
y B a
X
.
v
and Er i s thus the cotangent bundle! In general , if E is a vector bundle and r* is the representation r * (g) = (g - I ) T, then the associated vector bundle is called the dual bundle to E. Example 2: Let E again be the tangent bundle to Mil . Let G = Gl(n) act on mixed second order tensors JR." ®JR./' as follows. Let r : JR./l � JR./l be the standard representation r (g ) (v) i = g i j V j and let r * : JR./l. � JR.'" be the dual representation r * (g) (a ) i == a (g - I ) j i given i n E x a m p l e 1 . Then G acts on mixed tensors, say v ® a in ]R" ® JRn" j by the tensor product representation r ® r *
( r ® r * ) (g) (v ® a )
:
= r (g ) (v) ® r * (g ) (a )
483
ASSOCIATED BUNDLES AND CONNECTIONS
where v
=
Br vr and Ci
=
as dxs . The resulting bundle
of mixed second-rank tensors on Mil .
Er0r•
is then the familiar bundle
In a similar manner, essentially all the tensor fields considered previously were sections of vector bundles that were
asso ciated to the tangent bundle through some tensor product representation of the stru cture group of the tangent bundle or its dual!
IS.2b. Connections in Associated Bundles A con n ection in a vector bundle E assigns to the patch U of Mil a � , -valued I -form
{J)u ,
that is, a matrix of I -forms. This matrix acts on a section, given by the K-tuple a K-tuple of i -forms ding iel y
( wy ) R
=
w R s yS
=
y,
wJsy s dx j
We then have the covariant differential
'\Ju Yu = d y u + Wu Yu
( 1 8 .22)
and in each overlap
'\J v y v
=
c vu '\Ju Yu
Suppose now that we have a representation p : G
�
GI (N) of the structure group of
E . Since W is the tangent space to the manifold G at e and �,£ (N) is the tangent space to Gl(N) at p ee) = I , the differential p yields a linear transformation p : �' � �,£ (N) . If
*
*
S E W' the I -parameter subgroup generated by S is exp(tS) . Since p is a homo morphism ,
the image curve p [exp(tS) ] is a I -parameter subgroup of GI ( N ) , and so is again of the form exp(tY) for some Y E W£ (N ) . But the tangent to p [exp( t S)] at I is, by the definition of the differential, simply p * (S) , and so Y = p * (S) and p [exp( t S)] = exp[ t p* (S) ] ( 1 8.23) For example, in the homomorphism p : U ( 1 ) � GI(2, q given by p (e i 8 ) = diag(ei e , e 3 i 8 ) , i E lL( 1 ) gets sent into the 2 x 2 matrix p* (i ) = diag(i , 3 i ) . In the homomorphism g � p eg) = r * (g) = (g - I ) T o f G into itself, exp (tS) gets sent into exp( -tST), and so p * (S) = _S T . Let now E p be the bundle associated to E through a representation p : G � GI(N) . We define an associated connection for E p by using as connection form in U
( 1 8.24) which is defined as follows. Let X be a tangent vector to Mil . Then W u (X) P. [wu (X) ] E p * ( W) = �l (N) . Then we define In particular
[p* wu ] (X) : = p * [wu (X) ]
E
� ' and so
( 1 8 .25) ( 1 8.26)
484
CONNECTIONS AND ASSOCIATED BUNDLES
(18.27): {Qu } defines a connection/or the bundle Ep associated to E via the representation p .
Th eorem
Before looking at the proof we consider two examples. Let w be the connection form for the tangent bundle E = T M il . Example 1': We have seen in Example 1 that the co tangent bundle is associated with the representation r * ( g) = (g- I ) T . We have also seen that p * ( S) = _ST for all S E �. Hence Q = _w T is the connection form for the cotangent bundle, that is, Q j - ( rj l . =
Thus for covariant derivative w e get
which agrees with Equation ( 1 1 . 1 2). Example 2' : As in Example 2, consider the vector bundle of mixed second-order tensors * associated to the tangent bundle through the representation p r 0 r . For any 1 parameter subgroup g e1s o f G w e have =
=
p (exp and thus
p* ( S) (v 0 a )
=
(S) (v 0 a)
= (exp
tS v ) 0 (exp -tST a)
d (exp (S) (v 0 0')]1= 0 d t [p
=
(Sv) 0 a - v 0 ( S T a)
We may write
and then ( 1 8.28) Thus
Afjj
=
Bj Af + r fK Ar - rfsA�
which is the familiar rule ( 1 1 . 1 3 ) for the covariant derivative of a mixed tensor. PROOF OF THEOREM
( 1 8 . 2 7 ) : Let us put P u v : = p (cu v )
for the transition matrices of the new bundle. We must show
Now
Q v (X)
=
pAwv (X ) ]
=
p* [cu � wu (X)cuv + cu �dcu v (X)]
Consider the two terms on the right-hand side. For brevity, let us write
W u instead of W u (X)
485
ASSOCIATED BUNDLES AND CONNECTIONS
Now W u E W is the tangent vector at e E G to a I -parameter subgroup g et) : = ex p (tw u) . From the geometric meaning o f the differential (and using the fact that we are at a fixed point x E U)
p * (c u �wu c u v )
=
d [p (c u �g (t)cu v ) ] t=o dt
which, since p is a homomorphism,
d _I [ (c u v ) p{g (t) } p (c u v )] t =o dt P d _ = p I (c u v ) [ {g (t ) }] t =o P (c u v ) dt P = p - I (c u v )p * (w u ) p (cu v ) = Pu � Qu P u v Consider now the second term p* (cu �dc u v ) . Let x = x (t) be a curve o n M having X as tangent vector at t = O. We then have a curve in the Lie group G =
cu � (x (O) )c u v (x (t)) that starts at the identity with tangent cu �dcu v C X ) d p. [cu �dcu v (X) ] = P [ c u � (O)cu v (x (t) )] t =o dt d = Pu � (O) [ u v (x (t) )] t = o = Pu �dp u v (X) dt P and we are finished.
0
From Theorem ( 1 8.27) we have that the covariant differential for the associated bundle is then
( 1 8 .29) and automatically
'1 V o/ v = p (c vu ) '1 u o/ u For covariant derivative
( 1 8 .30) If we do not suppress the fiber indices
'1j o/ R
=
8j o/ R + (p* wj l s o/ s
IS.2e. The Associated Ad Bundle
We may let G act as a group of linear transformations on its own Lie algebra by for all Y E �/ . Thus
( 1 8.3 1 )
486
CONNECTIONS AND ASSOCIATED BUNDLES
and one checks immediately that this is a representation of G , called the adj oint representation. The subgroup Ad(G) c GI( �) is called the adjoint group of G . For example, when G is abelian (e.g., the n-torus), Ad(G) reduces to the sing le identity transformation; this follows immediately upon differentiating with resp ect to t the relation g e lY g - l = elY. In Chapter 1 9 we shall see that Ad(S U (2)) is isomo rph ic to the group SO (3) . S ince Ad : G ---* GI( �1' its differential a t the identity takes � into the tangent space at 0 to the vector space �" , �l (�1 ' that is, all linear transformations of �
Ad. : � ---* linear transformations of �' into itself
and we can compute this as follows. Take the curve ( I -parameter subgroup of G) g ( t ) = elx starting at the identity of G . This yields the I -parameter group o f linear transformations o f � given by
Ad e l x (y)
=
e l x Y e -l x
The tangent vector to this curve in � , at t = 0, is, when translated to 0,
a d X(Y): = [X.Y] is the translate of ;)\- et xYe- t X l t = o
t o the origin 0 of 3
Figure 1 8. 1
Let us write ad (X) for the linear transformation �/ ---* �/ given b y Y ---* [ X, YJ . Thus Ad. (X) = ad (X) = [X, J Recall that a I -parameter group h (t) has an infinitesimal generator S such that h (t) = elS, and that S = h' (O) . Thus we have shown that for fixed X, the I -parameter
487
ASSOC IATED BUNDLES AND CONNECTIONS
X group Ade1 has infinitesimal generator S = ad(X) . I n summary ad (X) ad (X)Y We can then write
Ad(e 1 x )y
=
=
:
:
= =
Ad* (X) [X, Y]
( 1 8 .3 2)
[ I + tad (X) + � >d(X)ad(X) +
...
t2 Y + t [X, Y] + 2 ! [X, [X, V] ] + . . .
]y
( 1 8.33)
Returning to a vector bundle E with structure group G and connection w, we can w no consider the bundle associated to E through the adjoint representation Ad : G ---+ Gl( �) ; Ad(G) acts on the new fiber �/ . Then if E had transition functions C u v , the Ad (G ) bundle has transition matrices Adc u v : �' ---+ �
Adc u v (x ) Y
=
c u v (x) Yc v u (x)
and connection
( 1 8 . 34)
Then the covariant differential and derivatives are
V'y = dy + [w, y] V'j y = aj y + [Wj ' y]
( 1 8 .3 5)
where y = {y u } is a section of the Ad bundle E A d ; that is, each Yu (x)
Yv (x )
=
c v u (x) Yu (x)c u v (x)
E
�/ and
P roblems
1 8 . 2(1 )
The cotangent bundle T* Mn has transition functions G v u = ( 8 XU/8XV) T in G = G I (n; lR) . G acts on the 1 -di mensional vector space lR via the determ inant representation det; g E G sends r E lR to det ( g) r. One may then consider the rea/ line bundle, the determ i nant line bundle, associated to T* M via det.
(i)
Show that any globally defined exterior n-form on a cross section of this new bundle.
Mn can be used to define
(ii) If w is a connection form for the tangent bundle TMn (for example, the Levi Civita connection for a Riemmanian M) show that -trw is the associated connection for the determinant bundle and thus the covariant derivative of a section r/> is given by
488
CONNECTIONS AND ASSOCIATED BUNDLES
(iii)
1 8.2(2)
I f Mn i s Riemannian , the volume n-form vol n = ,J9dx i s a pse udofo r m . Th vol u me bundle is the line bundle with transition fu nctions c 'vu == I e I I d et ( a xu / a x v ) l · { ,J9 u l defines a global section of this bun dle. Sh o h -trw is again a connection for this bundle and show that the secti o n (v'§ a 1 defined by the volume form is covariant constant! This is the inte r pretati n of Equation ( 1 1 .28) !
;� �� �
[
]
The tangent bundle to an orientable su rface has transition functions Cu v =
CaS e sin e
- sin e cos e
when orthonormal frames are employed. Consider the representat i o n p : S O(2)
=
-+
U(1 )
defined by p (cu v ) e ifl . This defi nes an associated bundle; it is simply the tangent bundle considered as a complex line bundle. If (w jk ) is the <'> 0(2) matrix of connection forms, show that iW21 is the connection for the associated line bundle. This agrees with ( 1 6 .29).
1 8.3. r-Form Sections of a Vector Bundle : Curvature
Where do the curvature forms live?
18.3a. r-Form Sections of E In this section we generalize the notion of a (tangent) vector-valued r-form that played such an important role in Cartan's method in Section 9.3 and following. An r-form section of a vector bundle E over Mn is by definition a collection of r-forms {
that is,
=
c v u
( 1 8.36)
[
collection of r + I forms that is,
=
( 1 8.37 )
489
r -FORM SECTIONS OF A VECTOR BUNDLE: CURVATURE =
(Recall that W j
W (8 j )
E
�/ i s a matrix)_ Note that this merely says (
'17
R R =d R + w S
/\
s
( 1 8 .38)
It follows, as usual, that V is an (r + I )-form section of E , that is, 'I7 v = c v u 'l7 u
that is, we have covariance. Note that W is the connection for E, not
TM !
18.3b. Curvature and the Ad Bundle
We know that the local curvature forms e u = dw u + � [ wu , w u ] of the vector bundle E are �rvalued 2-forms, that is, matrices of forms. These local �rvalued forms, however, do not fit together to yield a global form; rather, they transform as e v = c v u eu c v � e v = A d ( c v u ) tlu
( 1 8.39)
Thus
Theorem (18.40): The collection of local curvature forms { eu } fit together to
give a global 2-farm section afthe Ad(G) bundle!
(To exhibit curvature as an N -tuple rather than a matrix, one introduces a basis {E R } of the Lie algebra and writes e = 2:: E R e R _ )
Consider the exterior covariant differential o f e in the Ad bundle associated to E . LeU = (i l < i 2 ) , K = (k l k2 k3 ) ' Then el E �/ and w e have from
( 'I7 e ) K
=
.I
I
8� { aj el + A d* wj (el ) } = 8� { aj el + [ Wj , el ] }
=
deK
+ [w , e l K
( 1 8.5') . Thus 'I7e
and since e
=
dW +
=
de + [ w , e]
Hw , w]
=
= dW + W 'I7e
=
de /\
+ W /\ e
- ( - 1 ) 2e
/\
W
W , we have again
0 (Bianchi identity)
( 1 8.4 1 )
In general, for any p-form section of the Ad (G) bundle
'17 F P
=
d F l'
+ [w , F P ] =
d F P + W /\
FP - (
-
I ) P FP
/\
W
A p-form section of the Ad bundle will be said to be a p-form of type
( 1 8.42) Ad(G ) .
Physicists traditionally d o not deal with exterior forms and thus they are forced to exhibit the space-time tensor indices. On the other hand, they usually suppress the Lie algebra index. For a I -form F l they would write ('17 F l ) jk
=
aj Fk - ak Fj
('17 F2) ijk
=
a; Fjk
+ [Wj ,
Fd - [Wb Fd
( 1 8.43)
and for a 2-form ('17 F2 ) ij k = 8 Ut l ( ar F'1 + [W,. , Fs1 D , that is,
( 1 8.44)
+ ak Fij + aj Fki
+ [ Wi ,
Fjd + [Wb Fij ]
+ [ Wj ,
Fki ]
490
CONNECTIONS AND ASSOCIATED BUNDLES
P roblems
------
---
___
1 8.3(1 ) Show that for any p-form of type Ad( G) V 2 1/1
=
v V 1/I
=
(1 8.45)
[e , 1/11
1 8.3(2) Let ¢ P and 1/I q be form sections of an Ad(G) bundle, associate d to a vect
�
bundle with transition m atrice s cu v . Assume that G is a mat rix gro u p (i.e, , . subgrou p of G I ( N» . Si nce G .IS a subgro u p of GI(N), each C u v (x) is a matri in G I ( N) and we may th i n k of ¢ and 1/1 as fo rm sections of the A dG I ( N) bu n dl x Th i n k of them , as usual , as collections of locally defi ned matrice s { ¢u}, {1/1 } forms. Then ¢ u /\ 1/Iu is a local matrix of ( p + q) -forms, and tho ugh its va ues need not be in �" they will be in W(N) , which is sim ply the space of all N x N matrices. Show that ¢ /\ 1/1 is a ( p + q) -form section of the associated AdGI( N) bundle and show then that ( 1 8 A2) yields the Leibniz rule
Y
V (¢
/\
1/1 ) = ( V ¢ )
/\
1/1 + ( - 1 ) p¢
/\
( V 1/I)
I n particu lar, for any exte rior powe r of the curvature form v (e
/\
e
/\ . . . /\
e) =
�
(1 8.46)
0
1 8.3(3) Show that if ¢ is a p-form section of an Ad G bundle then tr¢ is an ordinary exterior p-form on M.
1 8.3(4) We have seen i n Section 1 7 . 1 c that given a constant 9
E G there is a right action of 9 on the pri ncipal bundle; locally it was defined i n "coordinates" by gu ---+ gu g, and then it was shown that this was com patible with the bu n dle structu re. One can not get a left action by this process; however, we can do the followi ng. Consider G-valued functions hu : U ---+ G on each trivialization patch U c M. Let hu act on n - 1 ( U) of the pri ncipal bundle P by
( 1 8 . 47)
Show that these local actions fit together to give a global transformation of i nto itself provided
P
( 1 8.48) Thus we have the fol lowi ng. Consider the fiber bundle associated to the prin cipal bundle P, whose fiber is again G but where G acts on itself not by left translatio n , but by the adjoint action, adjoint g : G ---+ G, of G on G (not on � adjointg ( g' ) := gg' g - 1
( 1 8.49)
This bu ndle is called the gauge bund l e . Thus the left action (18. 48) is globally defined provided { hu } defines a cross section of the gauge bundle. The left action is again called a gauge transformation, but we shall not discuss here the relation with the gauge transformations of Section g Ab. We do not claim that any such section other than h e exists fo r a given bundle P. =
CHAPTER
-
19
The Dirac Equation Spin is what makes the world go 'round.
1 9. 1 . The Groups
SO(3) and SU(2)
How does SU(2) act on its Lie algebra?
FOR physical and mathematical motivation for this section (which involves nonrel ativistic quantum mechanics) we refer the reader to some remarks of Feynman and of Weyl. Specifically, Feynman [FF, pp. 8, 9], in his section entitled "Degeneracy," shows that a process involving a specific choice of direction in space requires that the process be described not by a single wave function 0/ but rather by a multicomponent column vector of wave functions \II = (0/" . . . , o/N) T . He then indicates [pp. 9-1 2] , roughly speaking, that since the physics cannot depend o n the choice o f cartesian coordinates (x ' , x2, x3 ) of space, the N -tuples must transform under some represen tation p : S O (3) --* U (N ) of the rotation group S O (3) of space. This is not quite accurate; since eiy \II represents the same wave function (when y is a constant), p is only a "ray" representation, p (g)p (h) = e i y (g,h) p (gh) for a function y ( g , h) . Weyl [Wy, p. 1 83] shows that this can be made into a genuine representation, except that it is (perhaps) double-valued. We shall show in this section that there is a natural 2 : 1 homomorphism Jr of the special unitary group SU (2) onto S O (3) , thus yield ing a (perhaps double valued) representation of S U (2) into U (N) . An argument of Weyl [pp. 1 83-4] indicates that a multiple-valued representation of a simply-connected group is actually single-valued. We have seen in Section 1 2.2 that S O (3 ) is not sim ply connected, but we shall show in Section 1 9 . 1 c that SU (2) is simply connected, and thus the wave vectors \II transform via a true representation of the "covering group" SU (2) of S O (3 ) . The relationship with "spinors" will be discussed in Sec tion 19.2. A concrete physical example (the Stern-Gerlach experiment) is discussed by Feynm an in [F, L, S , vol . III, chap. 6] .
491
492
THE DIRAC EQUATION
19.1a. The Rotation Group SO(3) of IR3
]
Rotations of IR 3 about the z axis form a I -parameter subgroup - sin e cos e
0 0
o
for some E3
E
�A}-<3) . Then
E3
=
=
[
� eXP e E3 de
[
0
-1 0 0
1
0
]
0 0 0
1
]
=
e =o
=
exp e E3
� R (e ) de
]
e =o
and likewise for E I and E 2 • We use as a basis for ��3) EI
=
0 0
o
0 0 1
0
]
-1 , 0
E2 =
For Lie algebra, we compute
[ � � �] , 0
-1
0
E3 =
[! -� �l
( 1 9. 1 )
that is, [ Ei , E j ] =
L E ij k E k
( 1 9.2)
k
and then these are the structure constants of S O (3)
Consider now a I -parameter group of rotations with angular velocity w,
r
x
Figure 1 9. 1
Then
]dt
-
dr
1=0
=
w x r(O)
w
= dB /dt.
THE
GROUPS SO(3) AND SU(2)
493
On the other hand, this I -parameter subgroup is of the form R (t) = et S
for some skew symmetric matrix S, and so r t ) = R (t )r(O) = e t S r(O) dr = Sr(O) dt t =O
]
a nd
we conclude that
e
S (r) = w x r
Note in particular that the skew symmetric matrices E " E 2 , and E3 are simply the matrices of the linear transformations
Ej (r) = ej x (r) where {ej } is the standard basis of lR? Then we can write symbolically R (t) = exp( Ej u ) t ) = : exp(E . wt )
( 1 9 . 3)
in this case o f constant angular velocity. In terms of an angle of rotation, e = tde /dt , and the unit vector n along the axis w, R (e ) = exp(e E . n)
( 1 9.4)
represents a rotation through an angle e about an axis with unit normal n.
19.1b. SU(2) : The Lie Algebra ,:HL(2)
.uL(2) = � consists of skew hermitian matrices with trace O. Then i� is the vector space of hermitian matrices with trace to be considered as a real 3-dimensional vector space (i.e., it is closed under multiplication by real numbers). A basis for i� is given by the Pauli matrices -i ( 1 9.5) U2 =
0,
[0i 0 ] '
For example, exp(eu3 / i ) = exp diag ( - i e , i e ) = diag(e-ili , eili) describes a complete I-parameter subgroup of S U (2) for :s e :s 2rr . Note that the commutation relations are given by
0
( 1 9.6) which is the same as for SO (3) if one uses Uj /2i as new basis for � = �u(2) . We shall Soon see that SU (2) is simply connected. Lie group theory states that there is then a homomorphism from S U (2) onto S O (3 ) . (These groups are then locally "the same" : The proof is an application of the Frobenius theorem.) We shall exhibit the classical homomorphism Ad : S U (2)
-+
S O (3)
Thus we claim that the adjoint representation Ad (g)Y =gY g - l of S U (2) on its 3-dimensional Lie alge bra .llL( 2) yields (see Theorem ( 1 9. 1 2)) the standard representation of S 0 (3) on lR?
494
THE DIRAC EQUATION
We start out b y looking more carefully at the Lie algebra �.tI.(2) = � . � an d i to be considered as 3-dimensional vector spaces over real coefficients; i� ha s a given by the a 's and {aa / i } give a basis for �/. Define a map
* : ffi. 3
x*
=
X ·
are
gasi
---+
i�
R a = x aR =
X 1-+
[ z
x*
+ iy
x
x - iY -z
s
( 1 9 .7)
]
This linear transformation maps ffi.3 o n to the space of traceless hermitian matrices and has inverse given by ( 1 9.8)
Under the map *
el = ( 1 , 0, O) T
1-+
al
We shall use * to identify points points x of ffi.3 .
x*
in
i� ,
(i. e. , hermitian traceless matrices) with
From
and tr(aj ad = 0 if j =f:. k we see that if we define a real scalar product in i�, by :=
(h, h')
tr (hh')
then the Pauli matrices form an orthogonal basis (of lengths .J2) . Recall that every Lie group G acts on its Lie algebra � / by the adjoint action
Ad(g) (X)
=
( 1 9.9)
gXg- 1
for all X E �. Each Ad (g) is a linear transformation. In our case we consider instead the action of S U (2) on the hermitian traceless matrices i� / and we shall still call this Ad. Ad(u) is the linear transformation Ad (u) :
i�, ---+ i�
u E S U (2) sends x * E For each 2
x
i�J' into
ux * u - 1
( 19. 1 0)
2 u E S U (2) w e are associating a 3 x 3 matrix
Ad(u)
:
ffi. 3 ---+ ffi. 3
using the identification * of ( 1 9.7). Note that Ad is a representation of S U (2) by 3 matrices,
Ad(uu ' ) (x*)
=
uu ' x* (UU,) - 1
=
Ad(u) 0 Ad(u ' ) (x*)
x
3
THE
GROUPS SO(3)
AND
SU(2)
495
Note further that
(Ad(u)x" Ad(u)x, ) = tr( ux, u - I ux,u - I ) = tr(x,x,) = (x" x, ) and so
Ad is a representation of SU (2) by orthogonal 3 x 3 matrices. We claim that
these matrices also have determinant + 1 . To see this (and more) we shall discuss the topology of S U (2) .
19.1c. SU(2) I s Topologically the 3-Sphere The usual ("fundamental") representation of S U (2) is by 2 x 2 complex unitary matrices with unit determinant
C2
We shall show that S U (2) is topologically the 3-sphere S 3 . S 3 can be pictured as the � �4 set of unit vectors in Note that S U (2) : S 3 ---+ S 3 ; this is the meaning of being unitary. Note further that S U (2) acts transitively on S 3 , for ( I , 0) T E S 3 can be sent into a generic point (z I , Z2) T E S 3
by
u
=
[ZIZ -�2]Z1 2
E
(In fact, the second column is the unique vector in (z 1 , Z2 ) T and is such that det u = 1 .) From ( 1 7 . 1 0) we know that topologically S3
�
( 1 9. 1 1 )
SU (2)
C2
that is hermitian-orthogonal to
S U (2)
--
H
where H is the stability subgroup of the point ( 1 , 0) T . But, as we see in ( 1 9 . 1 1 ), H is simply the 2 x 2 identity matrix I. Thus S U (2)
�
S3
topologically. In fact we have seen that the correspondence S U (2)
simply by sending the matrix u into its first column u 1-+
(U I 1 '
U21
)
---+
S 3 is given
T
In particular S U (2) = S 3 is connected. Since Ad (u) is an orthogonal matrix, det Ad (u) is ± 1 . Since it is continuous in u and always ± 1 on the connected S 3 we see that the determinant is + 1 . Thus Ad (u) E S O (3)
Ad : SU (2)
---+ S O (3)
496
THE DIRAC EQUATION
19.1d. A d : SU(2)
�
SO(3) i n More Detail
: 5 U (2) ---+ 5 0 (3) given in (19. 1 0 ) is onto; that is, every rotation in JR. 3 is of the form (19. 1 0). Furthe rm ore, thISO representation is 2: 1 ; that is, for each rotation R there are exactly two matrices ±u E 5 U (2) such that Ad(±u) = R.
Theorem (19.12): The representation Ad
Let u (t ) be a I -parameter subgroup of 5 U (2) ; it is of the form u (t) ::: eth / i , where h is a hermitian 2 x 2 matrix. This produces a I -parameter subgroup of 5 0 (3) under our identification of i�" with JR. 3 (i.e., x. x) Adu (t)x '" Adu (t)x. e - it h x . e it h The velocity vector at x E JR. 3 is given by d d Adu (t)x. lt = o = / - i th x . e i th I t=O = -I. [h , x.] dt d i [h j aj , x k ad = -i h j x k [aj , ad = 2h J. x k t j k l al 2 (h x x) I af PRO O F :
�
=
=
-
�
The angular velocity vector of the I -parameter group Adu(t)x in JR. 3 is then w = 2 h , and from ( 1 9 .3 )
Ad exp We have just verified that
( T . t ) x. '" R (t) h
= exp(E . 2ht)x
( 1 9. 13)
( 1 9 . 14) and this is not very surprising considering the remarks after ( 1 9.6). For example, as we have seen, the vector h = (0, 0, l ) T , that is, h = a3 , generates the I -parameter subgroup of 5 U (2)
and this corresponds, under Ad, to the I -parameter subgroup of rotations of JR.3 (see Problem ( 1 5 .2( 1 ))) cxp 28 E,
�
[�
cxp 2
-2() 0 °
o
0
0
] [COS =
2() sin 2() °
- sin 2() cos 2()
o
0
°
1
]
Note that exp«()a3 / i ) describes a simple closed curve in 5 U (2) for ° ::: () ::: 2rr , and exp(2() E3 ) yields two full rotations in this same () range ! Since every rotation of JR. 3 is a rotation about some axis, that is, is of the form R, and A d i s R = exp (E . w() , we see from ( 1 9 . 1 3) that Ad exp ( a /2i o w() indeed onto. It is immediate that if Ad(u) = R then Ad(-u) = R also, so that the Ad representation is at least 2 : 1 ; that is, it is not faithful. It is an elementary result of group theory that =
497
HAMILTON, CLIFFORD, AND DIRAC
If t/J : G � G' i s a homomorphism of G onto G ' , then G' i s isomorphic to the coset (p- I (e ' ) is the kernel . space G / H , where H =
This is basically our "fundamental principle" ( 1 7 . 1 0) , for G acts on G' by (g, g ') J--+ ¢ (g )g ' and the stability subgroup of e' E G' is the kernel H = ¢ - I (e') . In our case we need to know that the kernel of the Ad homomorphism consists precisely of the two 2 x 2 matrices ± ! ; we will then know, from ( 1 7. 1 1 ), that sU (2 ) is a fiber bundle over S O (3) with fiber always consisting of exactly two points ± u . This should not surprise us since topologically S U (2) is S 3 , S O (3) 3 i s the projective space lRp 3 , and lRp 3 results from S by identifying pairs of antipodal points ! Look then for those special unitary u such that Ad(u) is the identity rotation I i n Il�? We thus need ux.u - = x . , for all hermitian x . with trace O. In particular aa- I U(Ja = u , for each Pauli matrix. Writing u in the form ( 1 9. 1 1 ) and putting a == 1 will show that Z I must be real. Putting ex = 3 will yield that Z 2 = O. Thus u must be of the form ± ! , as desired. D 1 9.2. Hamilton, Clifford, and Dirac
Why is it that a full rotation is something whereas two full rotations is nothing?
19.2a. Spinors and Rotations of lR3 We saw in the last section that there is a representation of S U (2) as a group of rotations of �3
Ad : S U (2)
exp
,
(�
. Ae
)
�
J--+
5 0 (3)
( 1 9. 1 5)
exp(E . Ae )
for any A = ( A I A 2 , A 3 ) , and that the mapping Ad is exactly 2 : 1 . Thus to a rotation Of �3 about an axis given by a unit vector A through an angle e radians one associates two 2 x 2 unitary matrices with determinant 1 ,
exp
[� ] . Ae
and
exp
[�
. A (e + 2 7T )
]
In other words, S O (3) not only has the usual representation by 3 x 3 matrices, it also x 2 matrices acting on ((: 2 . The complex vectors (0/ 1 , 0/2 ) T E ((: 2 on which 5 0 (3 ) acts in this double-valued way
has a double-valued representation by 2
are called (2-component) spinors. Mathematicians do not like double-valued any things; they prefer to say that S U (2) furnishes naturally a spinor representation of the 2-fold COver of 5 0 (3 ) . When S U (2) is thought of as the 2-fold cover of S O (3), it is called the spinor group Spin (3). The topological reason that 5 0 (3) can admit a nontrivial double-valued representa tion is that S O (3) is not simply connected. The reasoning is very much like that used
498
THE DIRAC EQUATION
i n complex function theory when showing that a region i n the complex pl ane sup P<>n ing a multiple-valued analytic function cannot be simply connected. The I _param eter subgroup of S O (3) - sin e cos e o
for 0 � e � 277: is a closed curve C in S O (3)
=
�O l
JR p 3 .
Figure 1 9.2
This curve can be deformed into the curve A of Section 1 3 .3b, Example (5), and this curve cannot be shrunk to a point. This subgroup is generated by E · A = E3 . It is covered in the group S U (2) by the portion of the I -parameter subgroup generated by (j
/2i
•
A
= a3 /2i
( 1 9. 16)
for 0 � e � 277: . This is not a closed curve in S U (2) since it starts at 1 and ends at - I . Of course, if we make 2 complete rotations in JR 3 , this I -parameter subgroup in SU (2) that covers it will be a closed curve on SU (2) = S 3 . This curve on S 3 can be shrunk to a point (why?), and by "projecting down" we can use this to shrink the curve representing 2 full rotations of JR 3 to a point. In this way one can distinguish between a full rotation (which of course brings every point of JR3 back to its original position) and two full rotations of JR3 about an axis!
This truly mysterious fact can be experienced on at least three different levels. We shall mention two manifestations here; after discussing the Dirac equation we shall discuss the significance for particle physics. The two remarks to follow are related to the topological fact that the closed c urve A i n S O (3) has the property that it cannot be shrunk to a point, whereas any even multiple of it can be.
HAMILTON, CLIFFORD, A N D DIRAC
499
1. Physio logically. Thi s is the old "waiter with a platter" trick ; see Feynman 's treatment in [F, W, p . 2 9 ] . 2. Mechanically. (This interpretation was given by Weyl.) We are going to show that the closed curve in 5 0 (3) described by rotating a rigid body twice about an axis through a given point 0 of the body can be deformed into the point curve representing no rotation at all. C onsider a mathematical cone in space, vertex at 0, with axis always the z -axis, and w ith (half) opening angle a. Consider another mathematical cone, congruent to the first, but this time fixed in the body with vertex at O . Move the body so that the body cone rolls around the space cone.
Figure 1 9.3 If the opening angle a is very small, then on looking down on the space cone one can see that when the body cone has come around to its original position, the body has made approximately two full revolutions about the z axi s, and as a tends to 0 the body rotation tends exactly to two revolutions. On the other hand, if we use an opening angle ex that is almost n /2, then the cones are very flat and the body cone will be seen to wobble, with hardly any rotation at all, and in the limit as a ---+ n /2 the body remains motionless ! Thus, when using a as a deformation parameter, the curve representing a rotation through 4n radians about the z axis (a = 0) can be deformed into the point curve representing no rotation (a n/2) . =
See also the picture in Wald's book [Wd, p. 346] . For an application to rotating electrical machinery you may read about an invention of D. Adams in the article [Sto] .
19.2b. Hamilton on Composing Two Rotations The relation ( 1 9. 1 5) is a powerful tool for investigating the product of two rotations. This is a consequence of the fact that that the Pauli matrices satisfy very simple product
relations
(0', ) 2 = ( 0'2 ) 2
( 1 9. 1 7 ) =
(O'3 ? = I
The infin itesimal generators Ej of S O (3) satisfy nothing like this; for example, ( E , ) 2 =
diag (O, - 1 , - 1 ) . From ( 1 9. 1 7) one gets not only the commutation relations ( 1 9.6) but also the anticommutator 2 x 2 matrices ( 1 9. 1 8)
500
THE DIRAC EQUATION
In Problem 1 9.2( 1 ) you are asked to use the commutation and anti-commu tati on for mulas to show the following. For any pair of vectors A , B i n JR.3
(CT . A) (CT . B) = (A . B) l + i CT . (A
x
B)
( 19. 19)
and if A is a unit vector
For unit A
(1 9. 20) corresponds, as we know from ( 1 9. 1 5), to a rotation R 1 of JR. 3 about the axis L ej A j through an angle of e radians. Let B be another unit vector with corresponding rotati on R 2 • Show
(� ) (� [ ( ) ] . [. 2
R I R2 = exp
. Ae exp
e
¢cos "2 cos "2
. B4»
( 1 9.21 )
. e .
¢ A ·B 1 sm "2 sm "2
]
e e e ¢ A + cos - sm . ¢ (A x B ) . ¢ B + sm . - sm - I CT ' sm - cos 2 2 2 2 2 This expression (via ( 1 9.20» exhibits explicitly the (cosine of) the rotational angle and then the axis for the rotation R 1 R2 . The expression ( 1 9.2 1 ) was known to Hamilton in terms of his quaternions rather
than Pauli matrices. We shall discuss the relation between these "algebras" next. For more information and nice pictures see the chapter on spinors in [M, T, W] . Finally note that we have mentioned before that the exponential map exp : -+ G W is onto in the case of a connected compact group such as S U (2) . Thus 0 9.20) sh ow s that every u E S U (2) can be written in the form
u = a l + i CT ' C
where a 2 + I C 11 2 = 1 . This expression is unique since a l is real and i CT ' C is skew hermitian.
19.2c. Clifford Algebras Let us abstract some of the properties of the Pauli matrices that will be important for generalizations. We shall be very informal. First note that 0" 1 , 0"2 , and 0"3 span a 3-dimensional vector space V 3 under addition and under multiplication by real scalars; V 3 is the space of trace-free hermitian matrices. In this vector space there is a quadratic form ( , ) given by half that in ( 1 9.9), that is, (h , h ' , ) = 0 /2) tr hh', and then
( O"j , O"d
=
gjk = O jk
Furthermore, there is a multiplication (in this case matrix multiplication) in V 3 but V 3 is not closed under this multiplication. For examples, 0" 1 0" 1 = 1 is not in V 3 and
HAMILTON. CLIFFORD. AND DIRAC
501
0'1 0'2 =
i 0'3 is not in V 3 (i is not real) . Suppose that we now try to "close" this system. We adjoin the new matrix e4 = 0') 0'2 and all its real multiples. Continuing, we define (in no particular order)
From the anticommutation relations ( 1 9. 1 8), we see, for example, that e) e2 = -e 2 e J , and so we needn't adjoin e 2 e ) . Note also that eg : = e ) e ) = I also follows directly from ( 1 9 . 1 8 ) . We may now form the real 8-dimensional vector space with basis given by , eg . From ( 1 9. 1 8) alone we see that this new real vector space is closed under el products (e.g., e7e) = e) e 2 e 3 e ) = -e l e 2 e l e3 = e] e) e 2 e3 = e2 e3 = es). Note further that in a monomial expression (such as e7e ] ) any repeated basis element of V 3 (such as el ) can be eliminated by using ( 1 9. 1 8), yielding an expression having two fewer basis elements. The vector space (of 2 x 2 matrices) of real linear combinations of these e's is 8-dimensional and forms, as can be verified, an associative algebra, that is, a vector space with a composition (called product) that is associative and is distributive with respect to addition. In fact, in this case, this 8-dimensional vector space is simply the algebra of all complex 2 x 2 matrices ! This algebra is generated by the Pauli matrices, and will be called the Pauli algebra. •
.
.
.
Definition (19.22): If Cn is an associative algebra (over JR) with "unit" I , gen erated by an n -dimensional vector subspace V", if ( , ) is any real quadratic form on V n , and if V" has a basis e] , . . . , ell satisfying ( 1 9.23) where g jk : = (ej , ek ) , then CIl = c ( vn ) is called the Clifford algebra generated by vn with the quadratic form (, ) . Note that we put no requirements on the quadratic form ( , ) , but of course the resulting Clifford algebra will depend on the choice of ( , ) . For example, consider a Clifford algebra generated by an n-dimensional vector space V" with quadratic form (, ) identically O. Then we have e j ek = -ek e) , for all j, k, and of course (e ) ) 2 = O. The resulting Clifford algebra is simply the exterior algebra based on the vector space V" ! In general, as a vector space, Cn is generated by expressions of the form ei e j . . . ek . Each (e ) ) 2 is a mUltiple g jj of the identity, and thus commutes with everything. Also, as we have seen, we needn't consider expressions containing a repeated basis vector ej . From ( 1 9. 1 8) we need only consider expressions e i ej . . . ek that are ordered, i < j < ' " < k. It is then obvious that as a vector space (i.e., neglecting the product structure), the Clifford algebra c ( V n ) is isomorphic to the exterior algebra A ( V n ) and thus has di mension 2" . For example, the Pauli algebra, as a vector space, is isomorphic to the exterior algebra on JR 3 with abstract basis given by 0' , 0'2 , and 0'3 , but of course the exterior product is 1 far different from the product of Pauli matrices, that is, the "Clifford" product.
502
THE DIRAC EQUATION
I n an exterior algebra, the real coefficients, that is, the scalars o r O-fonns , sp I -di mensional subspace. In a Clifford algebra, the scalar mUltiples of the "u n it" I a nn a I -dimensional subspace that can be identified with the coefficient field R To form a Clifford algebra with generators ej , . . . , ell and quadratic fonn ( , ) , Vi simply consider all "formal expressions," e i ej . . . ek with i < j < . . . < k, and impo e the "relations" ( 1 9. 1 8) . It can be shown that the result is indeed a Clifford algeb ra. us look at some examples. Co is the algebra over 1Ft with no other generators; thus there are no e j 'so Co ::: lR is simply the algebra of real numbers. Let V j be a I -dimensional vector space with basis e I , and quadratic form (e j , ej) :::: - 1 Form the 2-dimensional vector space with formal basis consisting of e j and a new vector "e j e j " satisfying ( 1 9.23), ( e I ) 2 = ( - 1 ) / . Thus we are adjoinin g to V j a I -dimensional vector space to accomodate the scalars (i.e., all real multiple s of - 1 ). The basis element e I will be called i , the element ( ed 2 will be identified with the real number - 1 , and the 2-dimensional vector space over lFt is simply the algebra of complex numbers a + bi, C j = 1::. Let V 2 be a real 2-dimensional vector space with basis e j , e 2 , and quadratic form (e j . ed = - O j k . We write ej = j , e2 = k. We adjoin a I -dimensional vector space to accommodate the scalars j 2 = k2 = ( - 1 ) l . We adj oin another I -dimensio nal vector space to house the new element i : = jk = -kj (from ( 1 9 . 1 8)). Then ijk = i 2 = jkjk ::: - jjkk = - / , which is not a new element. Thus we needn' t adjoin anything else. C2 is Hamilton 's 4-dimensional algebra of quaternions a + bi + cj + dk. Let V 3 be a 3-dimensional vector space with basis (}I . (}2 , (}3 and this time with scala. product ((}j ' (}k ) = + O j k . We have discussed this case previously. Adjoining products of pairs (}j (}k satisfying ( 1 9. 1 8) yields a I -dimensional space of scalars (e.g., ()? = /) and a 3-dimensional space spanned by (}j (}2 = - (}2 (} j , and so forth. Another I -dimensional vector space is adjoined to house i : = (} I (}2 (}3 . C3 is the Pauli algebra (but note our choice of scalar product) .
f�
r.:
.
19.2d. The Dirac Program: The Square Root of the d ' Alembertian We wish to emphasize that we are continuing to use our choice of metric in Minkowski
space,
that is, (g j k ) =
( 1]j k )
:=
diag(- l , + 1 , + 1 + 1 )
although most treatments of quantum mechanics use the negative of this form. Schrodinger's equation ( 1 6.40) treats time and space differently and is thus not relativistic . The first relativistic wave equation was proposed by Schrodinger, but w as abandoned by him. It was then reintroduced by Klein and Gordon and is now called the Klein-Gordon equation. For a particle of mass m it is
D'f'
. - g j k aj ak ,'f'I, - m 2 1/1
, /, . -
( 1 9. 24)
503
HAMILTON, CLIFFORD, AND DIRAC
!b at is ,
rj 2 1/r - a t2
+
a 2 1/r aX2
+
a 2 1/r a 2 1/r + 2 ay2 aZ
= m2 1/r
Dirac w anted to have an equation that was first order in t, as in the nonrelativistic Schrodin ger equation. Special relativity would then demand that it be first order in the atial variables x, y , and z. Thus, Dirac was led to construct a first-order differential
sp
operator
with some constant coefficients
? = y j aj y j such that
!bat i s , to construct a "square root" of the d' AIembertian. Then we could
solve the Klein Gordon equation by first solving Dirac's equation (using the physicist's convention of
pu tti ng n =
1)
( 1 9.25) Then as desired.
We then need
requiring that we put
+
y k y j = 2g jk = 2 1J jk that is, the y s cannot be scalars (y l y 2 = _ y 2 y l ) . The y 's appear to generate a Clifford algebra ! It is then clear from Dirac's equation ( 1 9.25) that the wave function 1/1 cannot be a single-component complex function since the Clifford numbers y j would then take the complex numbers aj 1/r into a Clifford number y j (aj 1/r ) that could not be equated with the complex number m 1/r . Somehow the Clifford numbers must act on the wave functions in a less trivial fashion. yj yk
'
For relativistic purposes, Dirac also wanted "covariance" under Lorentz transforma
tions. We now turn to these matters.
Problem
1 9.2(1 ) De rive ( 1 9. 1 9 , 20, and 2 1 ) . Let R1 be a rotation of :rr /2 about the let R 2 be a rotation of :rr / 2 about the y axis. Describe R1 R 2 .
z
axis, and
504
THE DIRAC EQUATION
1 9.3. The Dirac Algebra
What is the topology of the Lorentz group?
19.3a. The Lorentz Group Our treatment of Dirac 4-component spinors to follow owes much to Bleecker's book, Gauge Theory and Variational Principles [Bl] . Our metric is, however, of opposite
sign.
The Lorentz group is by definition the group of linear isometries of Minkowski space Mri
L = {real 4
x
4 matrices B I ( Bx , By) = (x , y) }
with metric (TJ j d = diag(- I , + 1 , + 1 , + 1 ) . In matrix notation,
(x , y) = x T TJ Y and then
requires
( 1 9.26) We see that det B = ± 1 . Let eo , e l , e2 , e3 be an orthonormal basis. Since (Beo, Beo) = a 1 and Be o = [B a , B l o , B 2 o , B 3 o f , we see that (B o o) 2 � 1 . Bleecker shows that L breaks up into 4 connected components (pieces) -
> 0 and B a a � 1 L - t : det B < 0 and B a a � 1 L+ + : det B > 0 and B O o :S 1 L - + : det B < 0 and B O o :S - 1
La = L+
t : det B
-
where La is the component holding the identity. This is clearly the component consisting of Lorentz transformations that preserve not only the orientation of Minkowski space a (det B > 0) but also the direction of time (B a > 0) . Thus the orientation of 3-space is also preserved. Consider the Lie algebra e of L . Write B = el S . Then (e l s f TJe l S = TJ and differentiating with respect to t and putting t = 0 yield S T TJ + TJS = O. Since TJ T = ry , this says (TJS) T = - TJS. This merely says that when we lower the upper index of S ,
by means of the Lorentz metric, the resulting covariant second-rank tensor is skew symmetric ! Thus dim L = dim S O (4) = 6.
505
THE DIRAC ALGEBRA
5 0 (3) is covered twice by 5 U (2) . We shall now indicate why L o is covered twice by 51(2, C), the complex 2 x 2 matrices with determinant + 1 (which, of course, is again 6 dimensional). Let
H (2, C)
:=
{ 2 x 2 matrices A I A t
b e the 4-dimensional vector space (over JR. ) of
:=
(AY
=
A}
2 x 2 hermitian matrices with no
requirement on the trace. For a basis for H (2, C) we augment the Pauli matrices by the unit matrix
TO
= (To : =
a = 1 , 2, 3
( 1 9.27)
x T T = x j Tj = X O TO + x · a
( 1 9.28)
I
Defi ne now a new map . : Mg ---+ H (2, C) by
x
E M
1--+
x.
:=
We can solve for x
( 1 9.29) Easily
( 1 9.30)
det x. = - (x , x )
We shall also have need for another identification of Mg with H (2, C), namely
Then
x' and one computes
=
[-XO
+z
x + iy
�
x iy -x - z
1
( 1 9.3 1 )
( 1 9.32)
det x' =
- (x , x ) x.x' = x'x. = (x , x ) I
The two maps . and * allow us to think of Minkowski space as being simply H (2, C) in two ways. By using . we have the following.
Theorem (19.33): The assignment to A E 51 (2, C) of the linear map Minkowski space
A (A) : Mri
=
H (2, C)
---+
A
: 51 (2,
C)
---+
Lo
of
H (2, C)
A (A ) (x). : = Ax.A t = Ax.A T yields a 2 : 1 homomorphism of 5/(2, C) onto Lo
A
( 1 9 . 3 4)
506
THE DIRAC EQUATION
?
Note that A i s similar to Ad ___�U (2) ---+ S O (3) ; i n fact w en A i s i n the subgroup SU (2) of SI (2, C) we have A T = A = A - I . Before proceedmg to the proof of ( 1 9.3 3) We shall investigate this similarity in more detail . For the notion of "deformation retract " see Section 1 5 .3d. '
Theorem (19.35): S U (2) is a deformation retract of SI(2, C) and S O (3) is a deformation retract of La. Proof sketch: A E S l (2, C) can be thought of as a pair of complex vectors [a l l , a 2 1 V and [a 1 2 , a 22 V spanning an "area" det A = 1 . By the usual Gram
Schmidt-like process used in Section 1 5 .3d in the case of SI (2, lR ) (but u sin g a hermitian scalar product instead) we may deform Sl (2, C) into its subgroup S U (2) , all the while keeping S U (2) pointwise fixed. S U (2) is thus a deformation retract of SI (2, C) .
For the Lorentz group we proceed as follows, using familiar facts about Lorentz trans formations. Consider the upper sheet H 3 of the "unit" hyperboloid in Minkowski space, -
x
5 + x-x = -1
--------��- x
Figure 1 9.4
Each Lorentz transformation in takes H into itself since Lorentz transformations preserve the Minkowski metric. By a suitable Lorentz transformation A E we may take the unit vector ( 1 O) T E H along the t axis into any other given vector (t xl of H , since any timelike vector can be along the t axis for some inertial observer. Thu s Lo acts transitively on H . The stability subgroup of ( I O) T is immediately determined to be
Lo
La,
[ 01 0 ] S O (3)
which we call I x S O (3 ) , or, more simply, S O (3 ) ; this is simply the subgroup of all spatial rotations of lR 3 in Minkowski space. Thus H is diffeomorphic to the coset space Lo/ S 0 (3 ) . In other words (see Theorem ( 1 7 . 1 1 » Lo is a principal fiber bundle over the base space H, with fiber S O (3 ) . Note that the upper hyperboloidal sheet H is diffeomorphic to lR 3 (under the proj ec tion (t , x) ---+ x» and so is contractible to a point. We now invoke the following
(0,
507
THE DIRAC ALGEBRA
Theorem (19.36): If En+ k is a bundle over a base space M il, if M is contractible to a point p E M, then E has the fiber over p as a deformation retract. In particular, S O (3) is a deformation retract of La . We shall not prove ( 1 9.36) here; a
detailed proof can be found in Steenrod's book [St] . The following picture, in the case at hand, makes it seem plausible.
Lo c' SO( 3)
Figure 1 9.5
Take a Riemannian metric for E. Each fiber is a submanifold of E . Consider the "horizontal" distribution b. of (n - k) -planes in E that are orthogonal to the fibers. M can be contracted to a point p . Let q ' be a point in E and let C be the curve swept out in M as q = Jr (q ' ) is deformed to p. There is apparently then a unique curve C' covering C, starting at q ' , tangent to b., and ending at some point in the fiber over p. (This is similar to the picture of parallel displacement described in Section 9.7b.) In this way, we deform E into Jr - 1 (p) . What is wrong with this sketch? We simply note that in the general case, if the distribution is not chosen with some care, that is, if the metric in E misbehaves, then the curve C' covering C may never reach the fiber Jr - 1 (p) .
b.
------+-
p=o
Figure 1 9.6
M
=
lE.
508
THE DIRAC EQUATION
For example, in the usual projection n : ]R2 ---+ ]R given b y (x , y) ---+ x , E :::: ]R2 is a bundle over M = R We have chosen a strange metric in ]R 2 and have indicated the integral curves of the "horizontal" distribution, that is, the orthogonal trajectories to the vertical fibers. The integral curve labled L is asymptotic to the y axis, and all the integral curves above L are also. The integral curves below L are bell-shaped, with the highest point of the bells tending to infinity as the integral curve is chosen clo ser and closer to L . For C we may take the interval [- 1 , 0 ] , ending at p = O. This clearly can be covered by arcs of the bell-shaped curves, but on the leaf L and above one will neVer reach the y axis ! Rather than use the subspaces orthogonal to the fibers, one should introduce a connection in the fiber bundle and then use parallel translation to Cover curves in the base space. The reader may consult [No, chap. 2] for details. This concludes our sketch of ( 1 9.35).
Corollary (19.37): Sl (2, q is both connected and simply connected, since SU(2) is. Lo is connected and each closed curve in Lo is homotopic to a curve in SO (3) representing a multiple of a full rotation in ]R3 about some axis, say the z axis. The even multiples are homotopic to a constant; the odd multiples are homotopic to a full rotation. PROOF OF
( 1 9 . 33 ) : First note that since det A
= 1,
(A (A)x , A (A ) x ) = -det { A (A)x} * = -det Ax * A t = -det A det x * det A t = - det x * = (x , x ) and so A (A) is a Lorentz transformation. det A (A) = ± 1 (since every Lorentz transformation preserves ± the volume form d x o /\ dx 1 /\ dx 2 /\ dx 3 ) . To show that the determinant is + 1 we need only know that Sl (2, q is connected, and this was proved in ( 1 9.37). Since A = I yields a Lorentz transformation I with B O o = 1 , connectedness of SI (2, q shows us that B O o ::: 1 for all A E S L ( 2, C) ; that is, A maps SI (2, q into Lo . It is immediate that A is a homomorphism, as you are asked to show in Problem 1 9 .3( 1 ) .
We must show that A maps SI (2, q onto Lo . First look a t the differential of A at the identity of SI (2, q . S E �2, q means that S is a complex 2 x 2 matrix with trace O. Then A * S is the linear transformation of Minkowski space corresponding to
x*
---+
d _ - [e I S x * ( e l s ) t ] 1 -0 dt
But this is simply Sx * + x * s t = Sx * + (Sx * ) t , that is, twice the hermitian part of Sx * . We claim that A * is 1 : 1 at I . Otherwise, for some S i- 0, Sx * is skew hermitian for all hermitian 2 x 2 matrices x * . Putting x * = I shows then that S would be skew hermitian, (i.e., that S E Ml(2) ) . Thus if A * S = 0 then S E 6-u,(2) . But A restricted to the subgroup S U (2) is a local diffeomorphism into S O (3) , as we have seen in Theorem ( 1 9 . 1 2) . Thus A * is 1 : 1 at I . It is not difficult to see
509
THE DIRAC A LGEBRA
th at the group property, that is, the fact that A is a homomorphism, would show that A * is 1 : 1 at all points of SI (2, q . Thus A is a local diffeomorphism near each point of S/ (2, q . We conclude that the image U : = A [S/ (2, q] is an open subgroup of La of the same dimension 6, since the image of an open set under a homeomophism is again open. But La is then the disjoint union of the open co sets of U . It is plausible, and can be proved (see [S]), that a space in which any two points can be connected by an are, say La, cannot be written as a disjoint u ni on of two or more open subsets. It must be that there is only one coset, that is, A [SI (2, q] = La, and thus A is onto. We need only show then that A is 2 : 1 . Ker A consists of those A E SI (2, q = I shows A t = A - I , that for all hermitian x * . Putting such that Ax * A t = is , A E S U (2) . B ut we have already seen in Theorem ( 1 9. 1 2) that A = ± I . Thus
x.
x.
L0 and we are finished.
_ SI (2, q - {±I}
0
A s SU (2) ---+ S O ( 3 ) yielded a double-valued spinor representation of the rotation A yields a double-valued spinor representation of the Lorentz group Lo. It is simply the usual representation of SI(2, C) as 2 x 2 matrices. This spinor representation of the Lorentz group La will be denoted by group, so
19.3b. The Dirac Algebra We h ave seen in Section 1 9.2 that the Pauli matrices (without ao ) generate a Clifford algebra C3
and that Dirac's program requires a C4 • There is a rather standard procedure leading
from a Cn to a Cn + 1 We shall only be concerned with going from the Pauli algebra to C4 • There is a complication due to the Pauli algebra using the metric 8af3 in ]R 3 while •
relativity requires that we use the Lorentz metric TJ jk in Mri. We proceed, with B leecker, as follows. I n the case of the Pauli algebra, the map * : ]R 3 ---+ 2 x 2 matrices can be thought of as a map a : ]R3 ---+ 3'1'(2, q a (x ) = x *
= (7' . x
For example, a ( 1 , 0, O) T = aJ , and so on. We now define a map : ]R4 ---+ q (i.e., all 4 x 4 complex matrices), by
y
�4,
[
0
y (x ) = x*
x* 0
]
(The meaning of this will be discussed in the next section. )
( 1 9.38)
510
THE DIRAC EQUATION
In particular
YI
:= y(ed = [ 0
( 1 9 .3 9)
al
y(e2 ) = [�2 Y3 := y(e3 ) = [�3 Yo := y(eo ) = [�/
Y2 : =
�] �] �]
the famous Dirac matrices. (This is one particular representation of the Dirac matrices. There are others in use.) The matrices generate a Clifford algebra. In fact we have
y
E
Y E
x Mg, Mg, we have y(x)y(y) y(y)y(x) 2(x, y)/
Theorem (19.40): For all
+
=
where ( , is the Lorentz metric.
)
Both sides of ( 1 9.40) are bilinear symmetric functions of any such function I we have
PROOF :
4 /(x, y) = I(x + y , x + y) - I(x - y, x and it is thus sufficient t o verify ( 1 9.40) when the arguments But
[
[
as desired.
-
y)
x and y are the same.
y(x)y(x) = x0* x0* ] x0* x0* ] = [ x*0x * x *0x* ] [ (X 'OX) / (x,0x) / ] = (x, x) [ 0/ 0/ ] =
0
P roblem
1 9.3(1 ) Show that A :
8/(2, C)
---+
Lo is a homomorphism .
x and y. For
T H E D I R A C O P E R A T O R fj I N M I N K O W S K I S P A C E
511
1 9.4. The Dirac Operator {iJ i n Minkowski Space
What
is
a Dirac spinor?
-
Warning: Our choice of metric signature has always been ( + ++) as this is most
convenient for discussing the geometry of general relativity. Approximately half of the physics books use this convention also in general relativity. Most physics books, however, when discussing (special) relativistic quantum mechanics, use the metric with - - ) . In particular, their d' Alembertian is the negative of ours. This signature (+ introduces the imaginary unit i into many equations. For example they would write the Dirac equation ( 1 9.48) below as i y j a{t/l = m l/! . There are so many different conventions in use for the Dirac matrices that we feel that this will not cause much more confusion than is already present in the literature. We are mainly concerned with the concepts involved in this subtle subject and feel that a change of signature at this time would only put an added burden on the reader.
-
19.4a. Dira c Spinors In the last section we exhibited the Dirac matrices y generating a Clifford algebra C4 , the Dirac algebra. The space «:4 on which these y 's operate will be the space of values of our wave functions, that is, a wave function l/! will be a column of4 complexfunctions. The Dirac algebra will allow us to construct a square root of the d' Alembertian, ?! = L yj aj . There is a serious problem remaining; we have constructed Yj by using a specific frame in Minkowski space. We shall choose Yj to be the same matrix in each frame a because there is no preferred frame in Mri . Since Yj is the same matrix in each frame a and since aj is frame-dependent, it is clear that fj = L yj aj would represent a different operator in each frame ! In order to avoid this the "functions " l/! on which � operate must themselves be made to be frame-dependent! Let us see how the l/! 's are to transform. We have defined the matrix y (X ) for each 4-tuple X by
y (X ) =
[0 0] X*
X*
and by definition of Yj
Consider a Lorentz transformation
x' The Lorentz transformation
=
A of Mri
( ax ) [h '
x =
AX
A will correspond, under A:
SI (2, C)
�
La
to two matrices ±A E Sf (2, C) ; pick one of them. By ( 1 9 . 34)
512
THE DIRAC EQUATION
Lemma (19.41): The 2
x
*
2 matrix associated to A (A ) (X) under is
t-I X* A - I Recall from ( 1 9.32) that X X* X*X* * - (X, X ) . Thus if X is not lightlike X * (X, X ) X,:- I A
A ( ) (X) *
A
=
=
PROOF:
(X, X ) / , and det
X . ::::
=
But if we prove ( 1 9.4 1 ) when X is not lightlike, it will follow for all X by continuity and the fact that any vector in the light cone is the limit of sp ace like vectors. Assume then that X ) # O. Then
(X, A ( A) ( X) * ( A ( A ) X A ( A ) X ) [ A ( A ) ( X) * r l = (X, X)[AX * A t rl (X, X)At-I X,:- I A -I A - I X* A - I ,
=
=
=
0
t
Theorem (19.42): Let p : 51 ( 2 , q � GI(4, Q be the representation of51(2, rq by 4 x 4 complex matrices defined by P (A) =
Then the Dirac matrices satisfy y (A (A)X)
=
[ � A ?- I ]
p (A ) y (X) p (A) - 1
( 1 9.43 )
(Note: X and A ( A ) X are the components of the same vector X in the two Lorentz coordinate systems e and e' = e A - I . ) PROOF :
y (A (A ) X )
=
=
[ A (A)X* (AO) X* ] [ t- �* A - AXo A t ] [ � A?-I ] [ ;* �* ] [ A; I 1t ] 0
A
I
A
=
I
= p (A)y (X) p( A - I )
=
p ( A ) y (X)p (A ) - 1
0
How do we interpret this result? If X is a tangent vector to Mg we may define the matrix y (X) = y ( X) by expressing X as a 4-tuple. This depends on the Lorentzian frame e in which X = e X is expressed. If, however, for each Lorentz transformation A of M(j we also make a change of frame in V = C 4 given by the change of basi s matrix p eA ) , then we see from ( 1 9.43) that y ( ) = y (X) is then a well- defined linear transformation y (X) : V � V that is independent of the Lorentz frame. This follows since the matrix B of a linear transformation changes under a change of frame p precisely by B 1-+ p Bp- l .
X
T H E D I R A C O P E R A T O R f) I N M I N K O W S K I S P A C E
513
Equation ( 1 9.43) is written in physics books as follows. Let A i j be the entries of the matrix A (A) . Then by our definitions
y (A (A)X) and
=
A' j X j Yi
yield from ( 1 9.43) ,
( 1 9.44) As mentioned before, the usual representation of S/ (2, C) by 2 x 2 matrices A is called the spinor representation when thought of as a two-valued representation of Lo and it is denoted then by ';Il ( 1 /2, 0) . The representation using A t- I instead of A is called the cospinor representation and is denoted by ';11 (0, 1 /2) . Two component spinors 1/IL , transforming under A , are also called left-handed, whereas two component cospinors 1/IR transforming under A t - I , are called right-handed. In order for Y (X) to be a well-defined linear transformation 1/1
E
V
f--+
Y (X) 1/1
E
V
1/1 = ( 1/IL , 1/I R ) T must be a 4-component spinor or Dirac spinor; that is, it must transform via the representation p in ( 1 9.42) 1/1
f--+ P ( A ) 1/1
for each Lorentz transformation A of M�. In summary
Corollary (19.45): A Lorentz transformation A : M� -+ M� must always be accompanied by a change of basis p eA) : c4 -+ C 4 (as given in (19.42)) in spinor space. Only then will y (X) act on Dirac spinors. The representation p of S/ (2, q is written ';11 ( 1 /2, 1 /2) and is the direct sum of �N l /2, 0) and �}(O, 1 /2) .
19.4b. The Dirac Operator Consider M� with a given Lorentzian coordinate system x . A "wave function" 1/1 will be a Dirac spinor, that is, a function on M� taking its values in C4 and transforming as in ( 1 9.45). In terms of a (two-component) left-handed spinor 1/I L and a right-handed spinor 1/IR 3 1/1 = ( 1/1 1 , 1/1 2 , 1/1 , 1/I 4 l = =
As usual we shall write
( 1/IL , 1/I R ) T
(1/Ii , 1/Ii , 1/Ik , 1/I�) T
514
THE D I R A C E Q U A T I O N
One verifies easily that these new y 's also satisfy the Clifford relations y j y k + y k y j = 2 g j k I = 2ry j k I
We define the Dirac operator � sending wave functions into wave functions by a 1jJ �1jJ : = y l _. ( 1 9 .46) axl where, as in ( 1 9.38),
[ ±�k � ]
Yk : = y (ek ) =
This defines � in terms of the Lorentzian coordinates x . What happens if we consider the same definition using a system x' = Ax = ( ax' /ax ) x ? Then
.
1jJ ' = p (A) 1jJ
.
where A is a constant matrix. We then have a 1jJ :1' ,1, ' k k a 1jJ ' Y' 'I' - YJ g 'j ax k - YJ g 'j P (A) ax k ' ' i a 1jJ ax = Yj g 'j k p (A ) ax 'k axi _
= Yj
_
(
)
. a 1jJ ax'j p (A) g " axi axr
which, from ( 1 9.44), yields
Then 1jJ ' = p (A ) 1jJ
=>
�' 1jJ ' = p (A ) �1jJ
( 1 9 .47)
shows that the Dirac operator � is a well-definedfirst-order differential operator on 4-component spinors of type ':)'( 1 /2, 1 /2) in Minkowski space !
From ( 1 9.39) and ( g j k ) = ( ry j k ) = diag [ - 1 , + 1 , + 1 , + 1 ] o Y = Finally
=
[ l ao +0 (1" · 8
[ 0 �/ ] +1
0 ]
- l ao + (1" 0 8
ya =
[� �]
T H E D I R A C O P E R A T O R I N C U R V E D S P A C E-T I M E
515
Thu s the Dirac equations ( 1 9.25) become the coupled system ( 1 9.48)
or
( - at + (1' . 8 ) Vt R = m VtL ( at + (1' . 8 ) 1ftL = m VtR
Note that for a massless particle these equations decouple and we can get by with a s ingle equation for a 2-component spinor Vt L of type '1' ( 1 /2, 0) , ( at + (1' . 8 ) VtL = o. This is Weyl's equation, which was found later to be an equation applicable to the
neutrino.
1 9.5. The Dirac Operator in Curved Space-Time
Does it make sense to say that a body, on returning from a long trip through the wormholes of space, has made an "odd number of full rotations"?
19.5a. The Spinor Bundle Consider now a pseudo-Riemannian 4-manifold M 4 rather than Minkowski space. We suppose that there are patches { U, V, . . . } on M 4 and orthonormal frame ("vierbein") fields eu , e v , . . . on each. Thus
and in an overlap we shall assume
ev (x) = eu (x )c u v (x)
where C u v : U n V --* L a . (Recall that this i s only one of the four components of the full Lorentz group; we are assuming that M 4 is both space- and time-"orientable"). We shall need to construct some analogue of the space of 4 component spinors. In our
discussion in Mri of the Dirac spinors, we associated with a Lorentz transformation A the matrix A , one of the two 2 x 2 matrices of 51 (2, q covering 1\ . There was no problem in doing this since we were dealing with a single constant matrix 1\. Now, however, we shall have to choose for each I\ (x ) = c u v (x ) a matrix A (x) = c� v (x ) in SI (2, q from among the two ±A (x) covering it, and w e shall have to d o this i n a continuous fashion. The transition functions Cu v (x) for the tangent bundle certainly satisfy the requirement ( 1 6.3), but it is not at all clear that the c� v (x) can be chosen consistently to satisfy it because of the ambiguity ± A . If this can b e done, then w e say that w e have "lifted" the structure group o f the tangent bundle of M 4 from the Lorentz group to the group 51 (2, q and that M 4 has a
spin structure.
This would have the following consequence. Let M4 be a pseudo-Riemannian manifold that is both space- and time-orientable; We may then assume that the tangent bundle has structure group La . Let e and f be
516
THE D I R A C E Q U A T I O N
frames a t a given point p. Then there i s a unique A E La such that f == eA . If f(t ) . a I -parameter family of frames at p such that f(O) = f( 1 ) = e, then f( t ) == f(O ) 1\ IS yields a closed curve t f-+ A (t) in La starting and ending at I . S1 (2, q is a 2 -fold c (t) oVer of La, and thus this curve is covered by a unique curve t -+ A ( t ) in S/ (2, q startin � I . (Visualize this by analogy to SU (2) = S 3 , the 2-fold cover of S O (3) == IRp3 section 1 9.2a.) We know, from Corollary ( 1 9.37), that S L (2 , q , like SU ( 2 ) , i s s m pY connected, whereas La, like S O (3) , has the property that the closed curve t � 1\ (t ) . IS homotopic either to a full rotation about some axis, say the z axis, or to a con s tant ma The covering curve t -+ A ( t ) detects the difference; A ( 1 ) = I if t -+ A (t) descri an even number of full rotations, whereas A ( 1 ) = - I if t -+ A (t) describes an odd number of ful1 rotations. All this is for a I -parameter family of frames e(t) at a given point x . No spin structure is required. Suppose now that p is in a patch U covered by a Lorentzian frame field eu . (This patch need not be a coordinate frame.) Take the frame f(p) = eu (p) at p and transpon it arbitrarily but continuously around some closed curve C = C (t) , 0 :s t :s 1 , lyi ng in U , again returning to the same frame f(p). We can compare f(C (t» with f(p ) == f( C (O» as fol1ows. Identify all frames eu at points of U with the single frame eu at p. Then by comparing f(C (t)) with eu at C (t), f(C (t)) = eu (C (t)) A (t), we again trace out a closed curve t f-+ A (t) in La. The resulting curve in La can again be uniquely covered by a curve in Sl (2, q starting at I . In this way we may be tempted to say that if A ( 1 ) = - I then the frame has made an odd number of rotations, whereas if A ( l ) = I it has made an even number of rotations. Unfortunately this result might depend on the choice offrames eu in U ! To see this, consider a spa tia l example, rather than space-time, replacing La by S O (3) and SI(2, q by S U (2) . Let M 3 be the 3-torus T 3 , with angular coordinates x , y , z . Let U = T 3 and let eu be the frame a/ax , a/ay, and a/ az. Then with the preceding identification, the frame f = e u along the closed z-curve (0, 0 , z ) would make no rotation at all. We may consider a new frame field ev on V = T 3 defined by e v = eucu v , where
�
i �
�
Cu v (z)
=
[COS�
Z
Si z
- sin z cos z
o
0 0 I
]
This frame coincides with eu on z = 0 but rotates once about it as one moves along the z circuit. Clearly the frame f = eu along the z circuit now makes one complete rotation with respect to the ev frame, that is, by identifying frames in T by mean s of the ev frames. We see that the contradiction arises because the eu and e v frames are related by S O (3) transformations Cu v ; they are not related by SU (2) transformations . We cannot decide whether eu and e v at (0, 0, 0) = (0, 0, 2rr ) are related by the identity I in S U (2) or by I in S U (2 ) ! The same problem would arise in space-time. We also see that this problem in the patch would not arise if we restricted ourselves to frames in T that can be related by SL(2,q transformations, that is, by frames that "do not make full rotations about each other." If M 4 has a spin structure, that is, if Sl (2, q is the structure group, and if we transport a frame f around any closed path C in M 4 , returning to the same Lorentzian frame, then we can decide whether the frame has made an even or an odd number of complete rotations ! For we may consider the Sl (2, C) frame bundle to M, that i s, the -
T H E D I R A C O P E R A T O R I N C U R V E D S P A C E- T I M E
517
frame bundle but using the structure group SI (2, ((:) . The curve C in M is then covered a unique curve in this frame bundle, defined by f. Upon returning to the starting
by
point of
C, the lifted curve will return either to its starting point, corresponding to an
even num ber of rotations, or to a point in the frame bundle related to the initial point E SI (2, C), corresponding to an odd number of rotations. spatial toral illustration T 3 just considered, T 3 is covered by a single frame ur o In 3 fi el d e u , and this frame field does define a spin structure. T can also be covered by the 3 singl e frame field ev and so this also defines a spin structure on T , but it is a different 3 spin structure ! On the other hand, T does not admit any spin structure that includes both frame fields eu and e v , as we have seen; we cannot lift Cu v (z) uniquely to S U (2) by - I
for all 0 ::: z :::
2n .
This has the following remarkable physical manifestation: We assume that our space M does not admit a spin structure we will not to consider the Dirac equation). For example, we may assume that space-time le e ab b Minkowski space Mo . As we have seen in Corollary ( 1 9.45), the electron ply is sim wave "functions," 4-component Dirac spinors 1/1 defined over Mg , will be, in fact, cross sections of a bundle over M associated to the tangent bundle through the representation p of Theorem ( 1 9 .42). Thus the structure group of the wave function bundle is SI (2, C) , rathe r than Lo. These spinors will then have the property that a complete rotation of 1R3 will send a spinor 1/1 not into itself but rather to its negative - 1/1 . Aharonov and Susskind [A, S] have devised a hypothetical experiment illustrating this. Two cubical devices can theoretically be constructed so that when they are brought together and aligned at a common face, a current will flow from one to the other, and if the cubes are then separated slightly and one of the cubes is rotated through 2n about their common axis and then brought back in contact as before, current will again flow but in the reverse direction ! Even in the case of a general space-time M 4 with spin structure, the cubes can be separated, one of the cubes can be transported along any closed curve, and upon return the direction of the current flow will tell us of the number (modulo 2) of "rotations" made by the traveling cube ! The "obstruction" to having a spin structure can be measured by the cohomology groups of M, but we shall only remark that a spin structure exists if for example, H2 ( M ; 2 2 ) , the second homology group with 2 2 coefficients (see Section 1 3 .2), van ishes. Obstruction theory will be discussed more in Chapter 22. If M does have a spin structure, then we may replace the Lorentz structure group by SI(2, C) ; the fiber for the tangent bundle of M4 is still ffi.4 . (Recall that SI (2, C) acts on JR4 as follows. To a 4-tuple x one associates a 2 x 2 hermitian matrix x * = x°a-o + x . (T. Then A E SI (2, C) acts on x by sending x* to Ax * A - I and then reading off the 4-tuple that corresponds to this hermitian matrix.) If Cu v are the Lorentzian transition functions for the tangent bundle, we shall let c� v be the SI (2, C) transition functions . We then construct the new 4-component ti me M4 carries a spin structure (for if
Dirac spinor bundle :f, = �M Whose fiber is ((:4 and whose transition functions
Pu v : u n V
--+ GI (4, C)
518
THE DIRAC EQUATION
x
are given by
PU V ( ) =
I
p ( C U V (X » =
[ c�v(x)
( 1 9 .49)
0
(See the discussion following ( 1 6.3) for the construction of this bundle.) Th is Dirac spinor bundle � is simply the vector bundle associated to the SI (2, C) tangent b Undle via the representation p of Theorem ( 1 9.42) ! This spinor bundle is the bundle whose sections 1/1 will serve as wave "functions " on M 4 . From this point on we shall assume that M does admit a spin structure and that one has been chosen.
ai
The Dirac operator construction iI = y j in M6 will not work in our curved M4 ; in our proof that iI1 1/I' = p eA ) il1/I for Mo we used the fact that the matrices A E S/ (2 , C) were constant (global Lorentz transformations were used since Mo is covered by global by some sort of coordinate systems). We shall now have to replace = covariant derivative. The Riemannian connection on M 4 won ' t work because T M and 'iOM are different bundles. What we need is a connection in this bundle 'iOM th at is associated to T M through the double-valued representation p of Lo.
aJ 1/I a1/l/axj
19.5b. The Spin Connection in �M Let M 4 be a pseudo-Riemannian manifold with a Lorentzian connection. Thus for any tangent vector X to M 4 , Wu (X) E �, the Lie algebra of the Lorentz group. We are assuming that M 4 has a spin structure (we may then consider S/ (2, C) as the structure group of the tangent bundle) and we want a connection for the associated spin bundle �M of wave functions given by the Dirac spinor representation p
:
S/ (2, C)
--+
G/(4, C)
First we need to construct a connection for the tangent bundle whose structure group is S/ (2, C) rather than Lo. Let w be the connection form for the Lorentzian tangent bundle; this is simply the Levi-Civita or Christoffel connection. Since A : S/(2, C) -+ L o is a 2 : I cover, to Wu (X) E � there is a unique Wi (X) E 6�£( 2, C) such that A * w� (X) = Wu (X) (there are two "vectors" "above" Wu (X) but only one of them starts at I E S/ (2, C) . It is not difficult to see that the �Ji(2, C) -valued local I -forms w� so defined form the connection forms for the tangent bundle to M 4 whose structure group is SI (2, C ) . One only needs to show that A Aw'V (X) ] = A * [A - l w� (X) A
+ A - l d A (X) ]
1 . The proof is very similar to that in Theorem ( 1 8.27). Wi will be exhibited explicitly in ( 1 9.53). We now have a connection for the tangent bundle T M 4 with structure group SI (2. C) and we have a representation p of SI (2, C) given by 4 x 4 matrices. The Dirac 4component spinor bundle is associated to the tangent bundle through the representa tion p . The prescription for constructing the associated connection in �M is given by since A * is I
:
T H E D I R A C O P E R A T O R I N C U R V E D S P A C E -T I M E
519
( 1 8. 24). We need to find an Q such that
( 1 9.50) w hich is short for p * w ' (X) = Q (X) , where X is tangent to by an exp licit calculation.
M 4 at x. We shall exhibit Q
First we need to calculate A * : a£( 2, q --+ � , identifying the Lie algebra of SI (2, q .)/£( 2, q consists of al l 2 x 2 complex matrices z with writing trace O. By
w ith that of the Lorentz group .
as
a su m of a hermitian plus an antihermitian matrix, both with trace
0, we see that a £( basi s for � 2, q can be taken to be the aex 's divided by i and the aex 's, a = 1 , 2, 3. Since ia l = a2 a3 , and so on, and aex = aoaex. where ao = TO = I , we prefer to write
this basis as
( 1 9.5 1 ) Note that the first three give the standard basis for the SU (2) subgroup of S/ ( 2 q . The identity component of the Lorentz group is generated by rotations and "boosts." The infinitesimal rotations have a basis given by the matrices Eex of ( 1 9. 1 ), where a runs from 1 to 3, but augmented by zeros in the Oth row and Oth column. For our purposes it is preferable to introduce a minus sign in the Eex 's. The resulting x 4 matrix obtained from - E I will be called E2 3 , - E2 will yield E3 1 , and - E3 will yield E 1 2 . A boost in the 0 1 plane is given by the 2 x 2 matrix
4
augmented by 0 elsewhere. We then have as basis for � the matrices E2 3 , E3 1 , E12, EO I , E02 , E03 , where
E23
[
=
0
0
o 0
o o o
0 0 o 0 -1
Each
Eexp is a skew symmetric matrix and we shall define Epex : = - Eexp . The Eop are symmetric matrices and we define Epo : = Eop . The homomorphism A : S I (2 q --+ Lo i s given by [ A (A)x] * = Ax* A t , and so if
h E � 2, q ,
d [exp (th)x * exp(th t ) ] t=o dt We have essentially done this calculation for h = aex / i in ( 1 9. 1 3). We have, under A * , [ A * (h )x] * =
( 1 9.52)
520
THE DIRAC EQUATION
Let u s now calculate where A . sends aoa" = a" . Since a1
anticommutator d [exp (ta,,)x. exp (t a,, ) ] t =o dt
=
a" w e get no w
an
{ a" , x. }
o o = { a" , aox + a x f! } = 2 a" x + { a" , af! } x tJ f! o o = 2a"x + 2o" aox f! = 2a"x + 2aox " f!
For example, if a = 1 , A . a , is the infinitesimal Lorentz transformation th at sends (x o , x ' , x 2 , X 3 , )1 to (2x ' , 2x o , 0, O) T , and so A . aoal = 2 Eo , . ( 1 9. 5 3)
( 1 9.52) and ( 1 9.53) describe A . completely. Let W = (Wi j ) be the Levi-Civita connection for the pseudo-Rieman nian M4 , using an orthonormal frame e. Using W l o = WI O = -WO I = Wa h and so on, we h ave WO , 0 W0 2 W0 3 W=
WO I
0
W0 2 -W 1 2 W0 3 - w I J
In terms of the matrices E we have
W
=
W' 2 0 -
w2 J
W' 3 W2 3 0
L Eij wi j i <j
Now use w O {J = waf! , ( 1 9.52) and ( 1 9.53) to get
1 � j ai aj Wi ( 1 9.54) 2 L... .< . I J 51 (2, C) connection form w', whose values are trace-free 2 x 2
WI
and this exhibits the hermitian matrices. Now we must compute Q = P.w'. From
we see
for all h
E
�£( 2, q . Then Q=
1 2
-
� .. W 'J L... . . I <J
[
ai aj 0
=
-
521
T H E D I R A C OPERATOR I N C U R V E D SPACE-T I M E
B ut
[6 -�fJ ] Q = ( 1 /2) L fJ spin connection YO YfJ =
Ya YfJ =
[
;
aa fJ
�J
aa OfJ a YO YfJ W + 0 /2) L a < fJ Ya YfJw fJ . in the spinor bundle is given by
then shows that
Thus the
and
Q
I I k . - - Wj k Y j Y - w j k Y1 Yk _ 4 4 I = g Wj k [ y 1 , Y k ] _ -
( 1 9.55)
.
recalling that wij = - Wj i . The covariant derivative in the spinor bundle is then
( )
d* 1 dX . k = dt + 4 Wj k dt y 1 Y * dt and the curved Dirac operator applied to * is 1 j 1 j �,I, k ,l, Y i e i 'I' + 4 Wik Yj Y 'I' = '1' '1' + 4 Wik Y i Yj Y 'I' V*
[ ( ,1, )
Aj
k,, ]
( 1 9.56)
( 1 9.57)
In the presence of an electromagnetic field with covariant 4-vector potential =
A(ej ) , then as in ( 1 6.43) the flat Dirac operator � would be replaced by �-
() ie
h y1 Aj .
A and
CHAPTER
20
Yang-Mills Fields
20. 1 . Noether's Theorem for Internal Symmetries How do symmetries yield conservation laws'?
In Section 1 0.2 we discussed Hamilton's variational principle for a dynamical system consisting of a finite number of particles. We shall now consider variational problems associated with a continuum or "field." We are frequently concerned with a mUltiple integral variational problem roughly of the form
o
i Lo (x , ¢ , ¢x )dx o
;\
dx 1
;\
•
•
•
;\
dx "
=
0
where both the field ¢ and the domain of integration M might be varied; that is, we consider variations o¢ and ox . In physics, one calls a variation ox of the domain an external variation, whereas field variations are called internal. We have considered external variations when dealing with arc length (geodesics) and with area (minimal surfaces); in both cases we dealt with the variations directly, rather than writing down the Euler-Lagrange equations. In this section we shall investigate the tensor nature of internal variations in more detail and also the effect of such variations that leave the Lagrangian invariant. ¢ will usually be an N-tuple ¢a (t , x) = ¢" (x) of functions, that is, the local repre sentation of a section of some vector bundle E. In the case of a Dirac electron, we have seen that E is the bundle of complex 4-component Dirac spinors over a perhaps curved space-time. If E is not a trivi al bundle (or if we insist on using curvilinear coordinates) we shall have to deal with the fact that the derivatives a¢ a / ax) do not form a tensor.
20.1a. The Tensorial Nature of Lagrange's Equations Let M " + 1 be a (pseudo-) Riemannian manifold and let E be a vector bundle over M ; for definiteness we shall let the fiber be ]R N . A section o f this bundle over U C M i s des cribed b y N real-valued functions (¢� ) , where ¢v = c v u ¢u and c v u ex ) is a n N x N transition matrix function, c� u h ' A "Lagrangian" is a single "function" L o (x , ¢ , CPx ) of
523
524
YANG-MILLS FIELDS
x , and section ¢ , and (for our present purposes) its first derivative matrix ¢x
: ::::
d¢a f a
j
(We have given the local description of the section and its first derivatives in a p x . h U . Higher derivatives may occur, as they did in Hilbert's approach to relativ i � Section 1 1 .3 . In that case the bundle was the vector bundle of covariant s ymmetrjItl second-rank Lorentzian tensors ; that is, the sections ¢ were pseudo-Riem ann ian / tensors gij on M4 , and the Lagrangian L o was R l g I 1 2 , involving second derivatives the metrics.) We are concerned with the action integral
; metriC
;
S
=
i
L o (x , ¢ , ¢Jdx
/\ dX" . For this to be independent of coordin ates where dx = dxo /\ dx l /\ dx 2 /\ we shall assume that for each given ¢ , Lodx is a pseudo- (n + I ) -form on M. In term of the volume form y'gdx (for simplicity we omit the absolute value sign on g) we write Lodx = ,'{!o y'gdx , and so •
S=
•
•
i
�
·'{!o (x , ¢ , ¢J y'gdx
5i.'o is a true function or scalar, classically called the Lagrangian density. For the gravi tational field, Hilbert's ,'{!o is the scalar curvature R . We shall vary the section ¢ . We shall assume that the metric ofM a nd any connections used in E are not varied. We are interested in the first variation of the action, and we
shall use the same classical notation as we used in Section 1 0.2, but we shall emphasize here the tensorial nature of this process. First note that 52 is to be a scalar constructed out of first partial derivatives dA>a = j a¢" / ax of the section ¢ . The collections of partial derivatives aj ¢a do not form a tensorial object (for example, in the case when E = T M) and consequently it is not clear how one is to construct a scalar X'o ! Frequently, however, there will be a connection in the bundle E and then we can construct instead the covariant derivatives Vj ¢a = ¢/" , }
=
V¢"
__ .
ax }
These do fit together to form a I -form section of E (as described in Section 1 8.3), that is, a section of the bundle E ® T* M, a generalized tensor. (Note that a is an E index, not a T M index; thus it makes no sense to ask whether a i s a "contravariant" index since contravariant in our sense refers to the tangent bundle only ! ) There is then hope for constructing a scalar out of ¢afj . For example, suppose that the structure group of the bundle E is S O (N) and that the connection cv has its values in a�N ) ; that is, w is skew symmetric. Then if gij is the metric tensor for M we may form La ¢'jj ¢'jkgJk , and it is not difficult to see that this is indeed a scalar. This scalar could be written I V¢ 112 and might be called the square of the "gradient" of the section ¢ . Thus w e shall assume that E has a connection for the given structure group G, and that from ,'{!o we may form a new Lagrangian ,'t' constructed using covariant derivatives, ,,{!
= X'(x , ¢ , V¢) = ,'{! (x , ¢ , ¢/x ) = ,'{!o (x , ¢ , ¢.J
rather than partial derivatives. (This will not always be the case. In Hilbert's variational approach to relativity, the fields ¢ are the components of the metric tensor. 2 = R, the
525
NOETHER'S THEOREM FOR INTERNAL SYMMETRIES
scalar cu rvature, is expressible in tenns of partial derivatives of the metric tensor but not the covariant derivatives of the metric tensor, which are all identically O ! ) ( as i n Equation ( 1 0. 1 2)) and from the fact + WJb ¢h and 0j = From / = j t a connection W is assumed unaffected by a variation of we see immediately he th t a th t
8
¢ OJ¢Q
Then 8
oj8
8¢ ¢,
i ,f(x , ¢ , ¢ ) Jgdx i [ ( 0;: ) 8¢(l + ( O:�j ) 8 (¢j)] Jgdx /j
Now in an overlap
UnV
we have
oS.' (�) ¢t o¢
=
But ¢t = ct v
c¢�
shows that
=c
Hence if
¢
i s a section of the bundle
But
the contraction
8¢
is a section of
E
vc'
O 'f
=
bundle E*.
(20. 1 )
0
=
and so
then
( being basically
difference of sections) and so
a
(� o¢a ) 8 ¢a occurring in the first integrand of
(20. 1 )
is a scalar. Since
scalar by hypothesis, it must be that th e contraction
( ) 0 ,';.'
must also be a scalar. S ince
8 (¢/) ( o¢o,f ) -/ j
O ¢/ j
8
8 1M ,� (x , ¢,
¢jj ) .,(gdx is a
(¢ a ) /j
is a section of
E @ T* M,
defines a section of
it must be that
E* @ T M
(20.3)
Our usual rules of tensor analysis apply i n this situation. For example, we have the W for Then _ wT defines the connection for (see Example I ' following
connection
E.
Theorem
E*
( 1 8.27)). We have the standard Riemannian connections r and - rT for T M and T* M . Thus, as discussed in Problem I 6.3( 1 ), we have a connection in any tensor ) b defines a product of the bundles T M, T* M, E, E*, . For example,
.. .
(o,�)/ o¢/j 8¢
E* @ T M @ E ; for simplicity let us call it A �j . 1t is of the form A�j B� C b • Thus a is an E* index, b in an E index, and j is a T M index . Its covariant derivative, again written A �j k ' is a section of (E* @ T M @ E) @ T* M and would be given by sectio n of
Abj
a/k
" Abj - A bj w k c
Uk
We m ay invoke the Leibniz rule derivative of
B
a
(B� C b ) / k
c
=
a + Wkbc A cj
a
=
+ rj
k i A bi
a
Bl/ k ch + B� C7k ' where the covariant C i nvolves only w. Covariant
involves both w and r, whereas that of
differentiation commutes with contraction s , and so on.
526
Y A N G-M I L L S F I E L D S
We now proceed with our calculation o f the first variation. From (20 . 1 ) and 8 (q,a IJ) :::: we have
(8
( 85tj8 qlj) cJ(ti
is a section of E * ® T M ® E , and contraction yields that ( 8.�j a q,a. )8q,e Now is a section of T M , that is, is an ordinary contravariant vector field X j on M we may then write
)8¢a] )8¢a] jj = [( � [( � o¢/j 8¢(h div
If M is compact with boundary, we have
8
rJM 2(x , ¢ , ¢jj ) Jgdx JMr [( o¢O � ) ( OO¢�jj ) jj ] (8¢a )JgdX 8 + r [( : ) 8¢a ] Nj dS JaM 8¢jj =
_
�d
(20.S) (20.6)
where N is the unit normal to the boundary and d S = i N Jgdx is the n-dimensional area form. Thus if the first variation vanishes for all variations vanishing on o M, we have the (Euler-) Lagrange equations
8;� : = ( o8;� ) (o:�J jj
8
-
=0
(20.7)
where the left-hand side, called the functional or variational derivative, defines a section of E * . It is convenient to define div
( o�¢ ) ( o:�j ) jj :=
which is not a scalar but rather a section of E * . Without components, we may write (20.7) in the form
20.1h. Boundary Conditions
Suppose that ¢ satisfies Lagrange's equations. Then we see immediately from (20.6) that if we demand that = 0 on o M, then S = O. The condition
8¢
8¢
=
0
8
on o M
527
NOETHER'S THEOREM FOR INTERNAL SYMMETRIES
is call ed an
essential or imposed boundary condition; we simply prescribe the value
ofrP on the boundary. We see, however, that the boundary conditions
( )
- N} - 0 ,
- I, . . . , N a-
aX a cp'lJ
wil l al so yield o S = 0 when cp satisfies Lagrange's equations. These are called the natural boundary conditions. See Problem 20. 1 ( 1 ) at this time.
20.1c. Noether's Theorem for Internal Symmetries
Suppos e now that we have a I -parameter group of symmetries of the Lagrangian, that is, we suppos e that c({' is invariant under a I -parameter group offibe r motions cp � cp (a) . We aE shall mainly be interested in the case when there is a I -parameter subgroup g (a) = e
of the structure group, E E 3" and .�' is invariant under cp � g (a)cp. (In the case of the Dirac electron, we shall see that the Lagrangian is invariant under the U ( 1 ) action
Vr t-+ ei a 1/f on spinors 1/f .) In this case g is a matrix function g a h (a) of a. In a given local patch U of M, the section cp is represented by a column cpa and then the symmetry would aE be of the form cpa (a) = gUb (a)cp h = ( e ) a b CP h . Then ocpa = (acpajaa)a=O = Ea h cpa . The sy mmetry assumption yields o S = O. Thus if cp is a critical section, that is, if ¢ satisfies Lagrange 's equations, then for any compact submanifold M' of M with boundary a M' we have, from (20.3) and (20.4),
r )
M'
[( � ) ] /j Jgdx a c ocp u acp/j
Since M' is arbitrary, we conclude
=
r }
M'
div
[ ( � ) ] Jgdx ax ocp a acp/j
=
0
Noether's Theorem for Internal Symmetries (20.8): If cp satisfies Lagrange 's equations and if 8cp is a variation by symmetries of the Lagrangian, then
Corollary (20.9): For the i-parameter group ea E of symmetries we have
and thus the vectorfield J
has divergence O. We
shall mention an application of this to the Dirac equation in Section 20.2.
528
YANG-MILLS FIELDS
20.1d. Noether's Principle The principle behind Noether's theorem is of more applicability and importanc e than the specific formula given in (20.8). All internal first-variation problems lead to an expression of the form
8
J Ldx J [ 88cpL ] 8CPdX + f G (x , u
=
. U
u
cP ,
8cp)dx
(for all compact regions U C M) where the form of the functional derivative 8L/8rfJ depends on the number of derivatives cp/j , CP/j b . , appearing in L. A solution to the variational problem satisfies the Euler-Lagrange equations 8 L/8cp = O. If then we have a variation that leaves Iu Ldx invariant, that is, is a group of internal symmetries, then we must have G (x , cP , 8cp) = 0 for the solution cpo This identity can be called Noether' s theorem, and is frequently of the form div J = 0 for some vector field J. We shall illustrate this with the familiar cases of geodesics and minimal surfac es. A geodesic M I in W" is a solution to the variational problem
..
8
1 ( dX , dX ) I /2 dt M
�
�
dt dt
= 0
for variations 8x vanishing at any pair of prescribed endpoints p and q of M . (Thi s does not fit into the scheme of (20.8); e.g., M is the image in the n-dimensional W" of the unit t -interval ; furthermore, x takes the place of the field cP, but the x 's are local coordinates in the manifold W, which is not a vector bundle.) x satisfies the Euler equations VT/ds = O. Consider a vector field J on W that generates a I parame ter group of isometries (e.g., the rotations of the round 2-sphere W 2 ) . Such a field is called a Killing field, after the mathematician Killing, and its flow clearly leaves the Lagrangian [gjk (dx j /dt) (dx k /dt) ] I / 2 invariant. However, this "variation" 8x = J does not vanish at the endpoints. The first variation formula ( 1 0.4) has "boundary" terms, and yields, since VT/ds = 0, the result (8x, T) (p) = (8x, T) (q ) . S ince this holds for all p, q on M, we have
-
(8x, T) is constant along the solution M I
(20. 1 0)
and we can make this look more like (20.9) by saying
d (8x, T)
=
0
where d is the differential for the I -manifold M. Thus a Killing field 8x has constant scalar product with the unit tangent to any geodesic. See Problems 20. 1 (2 and 3) for some applications of this result. Consider now the generalization of a minimal surface in ]R3 . sub manifold of the Riemannian wn provided
8
fu vol'
=
Mr
is a minimal
0
for each compact region U of M and each variation 8x that vanishes on a u. We considered the case when M 2 is a surface in ]R 3 in Section 8.4, where we derived the
529
NOETH E R ' S THEOREM FOR INTERNAL SYMMETRIES
first variation formula of Gauss. We accept the higher-dimensional version of this in the form oX T vol�U l voIr = (20. 1 1 ) o ( H, OXN ) voir +
j
r
j
lav where H is a type of meaD curvature vector that is normal to M, O X T is the component of the variation vector ox along the unit outward-pointing normal D to a U that is tangent that is, O X T = (ox, D) . (For a derivation of (20. 1 1 ) see, e.g., [L] .) The mean to curvature H is more complicated than in the case of a surface in ]R 3 since the normal space to M is of dimension n - r rather than 1 , but we shall not be concerned with it at V
V
U,
thi s time. The boundary term, however, should be completely evident. The formula then says that a minimal submanifold M must have mean curvature H = O . For a minimal l M an d a general variation we have 0 Iv voIr = Ia u oX T vol�U l . Since VO I�U = in voIr
we can write this as o
j vol' lavr (ox, D in vol lavr r i ) vol' lav V
)
=
=
'
(OXT '
D ) in vol'
(OX T
=
where OXT is projection of ox tangent to M . We now apply Noether's principle; if ox is a Killing vector field on Wn , that is, the generator of isometries, then the tangential part of ox is a vector field on M whose M-divergence is 0
(20. 1 2)
In the next sections we shall give some physical applications.
Problems
20.1 (1 ) Let
p be a given function on a compact Riemannian man ifold with boundary. Consider the variational problem for a scalar function i5
i[gik rjJjjrjJ/ k rjJ
+ 2 p ] ,J9 d x
=
rjJ
0
Find the Euler-Lagrange equations and the essential and natu ral boundary conditions. These should all be expressed in fami liar, classical language.
20.1 (2) The flow generated by a Killing field X is a 1 -paramete r grou p
rjJt
gi
of isometries. Thus if Y and Z are fields that are invariant under the flow, ( Y, Z) = i y i Z i is independent of t along an orbit of X . (i) Show that in the Rieman n ian connection , Jacobi's equation of variation
(4. 1 0) can be written
dt VY
i can be chosen arbitrarily at a g iven point, jj j i 0
(ii) Show then that d( Y, Z) / d t =
since Y and
Z
= VyX
0 translates into ( Xii Xi
+ Xjf i ) y i Zi
=
0, and,
+ Xl =
These are Killing's equations, satisfied by every Killing vector field.
530
Y A N G-MILLS F I E L D S
(iii) Use these equations to show directly that (20. 1 0) holds.
(iv) Let p be a point at which II X 11 2 = ( X, X) achieves its max i m u m :f. O . Thus T ( X X) = 2 ( V' T X, X) = 0 for every vector T at p. Let T be t h e u nit tangent to a geodesic through p with arc length parameter s. Sh ow th at d2 ( X, x) / ds2 = 2 ( - R( X, T) T, X) + 2 ( V' T X, V' T X) , By considering (n - 1 ) such unit tangents Tex , which, together with X , are orthonormal at p, show ,
that
We conclude Nomizu's theorem: If M n h a s negative defi n ite Ricci cu rvatu re t h e n no K i l l i n g field
X =1= 0
we Bochner: A compact M with negative Ricci curvature has no nontrivial Killing vector field.
can ac h i eve its m axi m u m l e ngth
at any
point of M. I n partic u l ar,
have another theorem of
For example, the Killing field a/ax on the Poincare upper half plane (see Prob lem 1 0 . 1 (2» has a length that tends to infinity as we approach the x axis.
20.1 (3) Let the cu rve Y = y(x) i n the xy plane of ]R 3 be revolved about the x axis, yielding a surface of revolution M 2 . We may use x and the cylindrical angle 9 (the polar angle in the yz plane) as coordinates for M.
(i) Write down (using the picture) ds2 for this surface. Clearly J = alae is a Killing vector field on M 2 , since it generates the rotations about the x axis. Consider a geodesic C, e e (x) on M 2 and let O' (x) be the angle that this geodesic makes with the lines of latitude, that is, the cu rves.
e
=
(ii) Derive Clairaut's relation y cos 0'
=
constant along
C
Consider an infinite horn-shaped su rface of revolution given by y -00 < x < +00, where f is increasing, f' (x) > 0 for -00 < x < +00, and f(x) -7 0 as x -7 - 00 .
=
f (x ) ,
(iii) Show that a geodesic that crosses the latitude circle a t x 0 and i s not orthogonal to this circle will lie in the region x � _ a2 , for some a. What is the best value for a2 ? What happens in the region x > O ? =
20.1 (4) Geodesics in the Poincare upper half plane. The Poincare metric in M2 = { (x, Y) I Y > OJ is d s'2 = y- 2 { dx 2 + dy2 J . Since the metric coefficients g�� are independent of x, a/ax is a Killing vector field. Since dy2 / y2 ::: (dx2 + dy2 ) / y2 , the vertical lines x = constant are clearly minimizing geode sics. We are interested i n the other geodesics. Let T be the unit tangent to a geodesic and let 0' be the angle that T makes with a/ax, all in the Poincare metric. (i) Show that y- 1 cos 0'
=
constant k along the geodesic.
(ii) Show di rectly from the metric that a horizontal line cannot be local ly min i mizing , and hence is not a geodesic.
WEYL'S GAUGE INVARIANCE REVISITED
531
(iii) Show that if two Riemann ian metrics ds and ds' on a space are confor ).. 2 ds2 for some smooth function ).. , then ma l l y related, meaning ds'2 =
angles measured with ds coi ncide with angles measu red with ds'.
Since the Poincare metric is conformally related to the euclidean metric = d x 2 + d y2 , we see that the angle a in part (i) is the same as the euclidean angle. Use now the euclidean metric dSQ . But i n the euclidean metric dy/ dSQ = si n a along a cu rve.
ds2
(iv) Concl ude that da/dSQ
- k , and thus the geodesic has constant euclidean curvature, and is thus an arc of a circle (of perhaps infinite radius). Show that if the geodesic is not a vertical line, then k =I 0, and so it is not straight. Then at the highest point Yo , k = 1 / Yo . Show that the euclidean circle strikes the x axis orthogonally. Thus the geodesics of the Poincare metric =
are euclidean circles (or vertical lines) that meet the x axis orthogonally.
20.2. Weyl 's Gauge Invariance Revisited
What can global symmetries tell us about background fields?
We remind the reader that our formulas will differ sometimes by factors of i from those of most books since we are using the metric signature (- + + + ) .
We shall also use the physicist ' s convention of frequently putting 1i = 1
Our remarks about quantization, especially "second quantization," will be extremely brief and sketchy.
20.2a. The Dirac Lagrangian We shall exhibit a Lagrangian whose Euler equations are the Dirac equations for a free electron (i.e., an electron not interacting with any other field) in Minkowski space Mri. First we shall need to construct scalars out of 4-component spinors 0/ = ( 0/ " 0/2 , 0/3 , Vr4 ) T . Recall that 0/ t = 0/ T is the hermitian conjugate row matrix. Then 0/ t ¢ is a hermitian bilinear form that is invariant under unitary transformations of ((:4 , but, as we shall see, it is not invariant under the Dirac representation p ( A ) : C 4 -+ ((: 4 of ( 1 9.42)
that accompanies each Lorentz transformation A of Mri. We remedy this as follows.
Recall the Dirac matrices (with our choice of signature)
532
YANG-MILLS FIELDS
I t is clear that y a i s hermitian whereas yO is skew hermitian, and thus i y o is hermitian . We now define the Dirac conjugate spinor (or adjoint spinor) to 1/1 by (20 . 1 3 ) (The factor i appears because of our choice of signature.) Since i y o is a herm itian matrix, the bilinear form (20. 14) is again hermitian. This form, however, is not definite because of the switc hing components resulting from y O . We claim that
of
Theorem (20.15): The form 1/1¢ is invariant under the Dirac representation p. Thus it is a scalar under Lorentz transformations.
PROOF :
One sees immediately that (20. 1 6)
i
and so, abbreviating p e A ) to p , we have ( p 1/l ) t i y o ( p ¢ ) 1/1 t yO¢, as desired. 0 Since p t P =1= I , it is clear that 1/1 t ¢ is not Lorentz invariant. Since fl1/I is a Dirac spinor if 1/1 is (this is the content of (19.47)), we conclude � � Corollary (20.17): 1/Ifl1/I and 1/1 1/1 are Lorentzian scalars. For an electron of mass m we may try to form a Lagrangian by Vr fl1/l - m Vr 1/l . As we shall see, the first term needs to be made more symmetrical in 1/1 and Vr . The Dirac Lagrangian is defined by (20. 1 8)
where aj 1/l is really (aj 1/l )� = ( aj 1/l ) t i y o . We claim that the Euler equations for the Dirac action
1 !!.'e dx
yield the Dirac equations ( 1 9.48). First note that 1/1 consists of four complex fields 1/1) in Me: . Since these are complex, we may write them in terms of their real and im aginary parts, yielding eight real fields to be varied independently. It is simpler (and equivalent) to allow the eight complex fields 1/1 and 1/1 to be varied independently. These eight fields
533
WEYL'S GAUGE INVARIANCE REVISITED
compris e the section cp = (cpa ) appearing in (2004). In Problem 20.2( 1 ) you are asked to show that the Euler equations for the Dirac action yield the Dirac equations for 1/1 and the conjugate
(20. 1 9)
It is clear from (20. 1 7) that the Dirac Lagrangian is invariant under the I -parameter group of "gauge transformations"
(20.20) �
�
where a is any real constant. Under this variation 81/1 = i 1/1 and 8 1/1 = -i 1/1 . Noether's theorem (20.8) then shows that the 4-vector J defined by
Jk
:=
-ie � y k 1/l
(20.2 1 )
bas vanishing divergence i n Minkowski space (the electron charge - e i s put i n for future needs) provided that 1/1 is a solution to the Dirac equation. Thus for the spatial slice V 3 (t) we have
�dt }r
v et)
r}v(t) aatJ o dx /\ dy /\ dz = - r a, J Ol dx /\ dy /\ dz JV(t)
J O dx /\ dy /\ dz =
If we assume that the wave function
1/1 vanishes sufficiently rapidly at spatial infinity, the last integral vanishes by the divergence theorem and we have that
r
JV (t )
e1/l t (i y O ) 2 1/1vo 1 3 =
r
J V (t)
e1/l t 1/I vol 3
is constant in time. As we shall see in Section 20.2c, if we think of 1/1 as a classical (unquantized) field, this integral is interpreted as the electric charge, e1/l t 1/1 is the charge density, and then J k is interpreted as the electric current vector.
20.2h. Weyl's Gauge Invariance Revisited A guiding principle of Einstein's theories of relativity is that the laws of physics should
be expressed in a form that is independent of any particular coordinate system used. Let us first look at a simple example in Newtonian gravitation to see how coordinate changes can be used to infer information about interactions. Consider a "small" laboratory in free fall in our space, distant from any sizable bodies. With respect to a small cartesian coordinate system attached to the laboratory, a small test particle in free fall satisfies Newton's equations d 2 x/dt 2 = O. With respect to a second cartesian system that is moving uniformly with respect to the first, that is, x' = x - kt , where k is a constant, we again have the same Newtonian law d 2 x' / dt 2 = o. We may say that uniform translation is a symmetry of our system. Newton of course realized this. He maintained that there are distinguished coordinate systems in our universe, those that are at rest with respect to "absolute space" and those that are moving
534
Y A N G - M I L L S F I E I, D S
uniformly with respect t o it, and his laws hold in any such system. I f however w e allow k to vary in time, for example k = O j2)got, then x' = x - O j2)got 2 , an d Newton's equations become, in the new coordinate system, d 2 x' j dt 2 = - go. This additional term is simply telling us that our new coordinate system is accelerating with respect to Newton ' s absolute space. Bishop Berkeley and, later, Ernst Mach rejected the notion of absolute space; they would say that the new coordinate system was accelerating with respect to the bulk of matter in the universe, the distant matter in the universe, or as they would say, the "fixed stars," and this is the interpretation preferred today. The additional term, -go in this case, is informing us of the existence of the gravitational influence of the distant matter, even if we had been unaware of the notion of gravitation ! Even when the gravitational force vanishes, as it does for all intents and purposes in our free-fali laboratory located at a great distance from matter, the distant matter still inform s the laboratory, through gravitation, of which coordinate systems are to be considered as (approximately) inertial . I believe that if space were devoid of even this distant m atter, Newton 's laws would make no sense, since there would then be no intrinsic notion of an accelerating frame or that of an inertial frame. There would be no notion of the "mass" of a test particle, since mass is measured via accelerations. Newton 's laws of motion
are an indication of some "background field, " gravitation, that is interacting with the test particle, and presumably these laws need amending when this background field is
taken into account, particularly when the "strength" of the field does not vanish. We have learned of thi s background field through the fact that Newton's laws do not remain invariant under non-uniform changes of coordinates. Newtonian mechanics takes place not in matter-free space but rather space with a "uniform" distribution of distant matter. Similarly, the Minkowski space of special relativity is not general relativity with no matter present, but rather an approximation in general relativity ofa region in cu rved space far from a u n (fo nn distribution of distant ma tta
Consider the Dirac electron in Minkowski space Mg. A free electron is postulated to satisfy the Dirac equation (20. 1 9), derivable from the Lagrangian (20. 1 8). The Dirac equation may be thought of as a replacement for Newton's law. Both (20. 1 9) and (20. 18) are invariant under (global) Lorentz transformations of Mg, but not under more general space-time coordinate changes. To allow for the general coordinate changes we proceed as we did in Section 1 9.5; we change the Dirac equation by replacing the Dirac operator by introducing the Riemannian connection for true space-time and replacing partial derivatives by covariant derivatives, yielding the new Dirac operator ( 1 9.57)
1 . . k �l/f + 4 r / k Y ' Yj y l/f
The second term, involving r and l/f , is an interaction term, telling us how the gravi tational field interacts with the electron field.
20.2c. The Electromagnetic Lagrangian
Physicists, following Weyl in 1 929, have carried this principle a step further. For sim plicity we shall neglect the very small gravitational interaction, that is, we shall put
WEYL'S GAUGE INVARIANCE REVISITED
r
535
0, thus returning to the original Dirac equation (20. 1 9). Instead of considering chan ge of (space-time) coordinates x, we shall look at a change of the field (fiber) coordin ate 1jJ , that is, a gauge transformation. Although quantum mechanics assigns a physi cal, measurable meaning to each absolute value 1 1jJ" I , the argument or phase of 1/rQ == l 1jJ a l exp (i e,, ) , that is, ea , has no such meaning; one cannot measure the phase of a wave function or spinor. Both Dirac 's equation and his Lagrangian are invariant under the global gauge transformation 1jJ f-+ e i u 1jJ (e iu 1jJ I , e i u 1jJ 2 e i u 1jJ 3 , e iu 1jJ 4 ) T a
==
=
,
the term global meaning (in physics terminology) that a is a constant. This invariance is
cru cial since a global change of phase of all of the wave functions in quantum mechanics must leave the physics unchanged. A global gauge transformation is a symmetry of the Dirac equation. Since the phase of 1jJ is not measurable, we should be able to have invariance under a loca l gauge transformation, where a = a (x) varies with the space-time point x ! Clearly the D irac equation and Lagrangian are not invariant under such a substitution because of the appearance of terms involving da . It must be that there is some background field that is interacting with the electron. This background field will manifest itself through the appearance of a connection. Since each component 1jJ a of 1jJ is undergoing the same phase transformation, we shall forget the 4-component nature of 1jJ and simply write 1/1 � e iu (x) 1jJ. If we think of this as a change offrame in a complex line bundle with e i u (x) , then we need a connection in this line transition functions g� 1 = c v u (x ) bundle that transforms as W f-+ g- I wg + g- I dg = w + g- I dg = w - ida. If we define the real field, that is, I -form, A, by w = - i A , then A' = A + da (x ) . Thus our unknown background field A transforms in the same way as the vector potential in electromagnetism, suggesting (with hindsight) that we identify the background field with electromagnetism ! (Of course we could have written w = -ikA for any real constant k. Comparison with classical mechanics, as in Section 1 6.4, leads to the choice k e/h = e.) The new Dirac operator is then =
=
(20.22) If we now replace jJ by h in the Lagrangian (20. 1 8) we get a new Lagrangian, which now contains terms involving the field A.
- . I - [1jJ y l (aj - ieA j ) 1jJ - (aj + ieA j ) 1jJ yl 1jJ] - m 1jJ 1jJ 2 - . . 1 - . = - [ 1jJ y 1 aj 1jJ - (a j 1jJ ) yl 1jJ] - m 1jJ 1jJ - (ie)A j 1jJ yl 1jJ 2 since (aj - ieA j ) t = a) + ieA j ' Note that the last term is, from (20.2 1), Wj :;f y ) 1jJ = -ieA ) :;f y j 1jJ = A j ] ) -"..)e
=
(20.23)
Quantum mechanics then dictates that the A field is also to be considered as an
independent field in its own right; that is, we are also to allow variations of the new Lagrangian involving variations of A . To get nontrivial field equations for A we need to have "kinetic" terms, terms involving first derivatives of A with respect to t. To maintain
536
YANG-MILLS FIELDS
Lorentz invariance w e shall need all first derivatives aj A k in the Lagrangian. These partial derivatives do not yield a gauge covariant quantity; one cannot form a gauge invariant scalar for the Lagrangian simply by taking L;(aj Ad 2 . Geometry tells Us that the curvature e 2 = d w + w A w = dw, with components -ie(aj A k - ak A j ) = : - i e FJ.k , is the correct tensor to use, rather than aj A k • We then add some multiple of the squ are of this electromagnetic jield strength F 2 to the Lagrangian. Our choice of - ( 1 / 1 6Jr) for this multiple will be vindicated shortly. This is ourjinal Lagrangian. (20 .24) -
- m '/"/'
+
'i' 'i'
.
A }· ]1 -
l
F k Pk 1 6:rr }
·
-
Look now at the variational equations involving 8A. Note first that for variations 8 A vanishing outside a small region
8
J _1 6:rr1 - Fjk F jk voI4 8 J _8:rrl_ F =
=
Also
8
1 8 _ ( F, �
A
*F
F) =
1 8 _ (dA , dA) 8:rr
(d8 A , F)
(8 A , d * F)
4:rr
4:rr
J A j ]1 vol4 J 8 A J i vo 4 =
where ;;;3 : = iJ vol 4 . We conclude then that d * F ( 14. 1 2), and so we have d * F = 4:rr ;;; . Since d F
1 =
= =
=
(8dA , dA) 4:rr
(8 A , *", 3 )
4:rr *;;;3 . But d * F = *d*F, from 0, we conclude that variation of
the A jield yields Maxwell 's equations provided that we identify J 0/ (20. 20) with the electric current density. Charge conservation d;;; = 0 follows. In summary, the Dirac Lagrangian (20. 1 8) admits the global symmetry group (20.20). lf we insist that the Lagrangian should admit local symmetries, when a is not constant, then Weyl 's procedure leads to the introduction ofthe "electromagneticjield " A; Maxwell 's equations (and charge conservation) then follow!
20.2d. Quantization of th e A Field: Photons We have now a Lagrangian involving the two fields 1/f and A . Quantum mechanics then requires that these fields be quantized; that is, these fields in some sense are to be represented by operators and one performs "second quantization" (see, e.g., [Su, chap. 7]). The quanta of these fields, which automatically appear, are interpreted as particles associated with the fields. Very roughly we have the following. The 1/f field yields again the electron. The 1/f t field also yields a particle, the positron, which had been predicted earlier by Dirac j ust on the basis of his new equation. The "gauge field" A yields another "new" particle, the photon. Physicists then say that the electromagn etic
T H E Y A N G-MILLS N U C L E O N
537
fo rce between electrons is "explained" by the exchange of these new "gauge particles, " th e photons, between electrons.
We should also remark that in the process of quantization the current (20.2 1 ) gets repl aced by a new operator; in particular, the density becomes the electron-positron charge density, rather than simply the electron charge density. In a few sentences, the guiding principle, the gauge principle, for studying the force between particles can be stated as follows. If a proposed Lagrangian of some matter field 1/1 is invariant under global (but not local) gauge transformations, alter the Lagrangian by replacing partial derivatives by covariant derivatives (introducing a new gauge field, a connection w, or potential A, whose transformation rule is compatible with the gauge transformations) ; the Lagrangian then has local gauge invariance. Then add to the resulting Lagrangian a new term proportional to the square of the "length" of the curvature dw + 1 /2[w, w] of the gauge field (to be more fully explained in the next section) so that gauge invariance is not destroyed. Variations with respect to 1/1 yield the field equations for 1/1 and variations with respect to w yield the field equations for the gauge field. Then when one quantizes the gauge field, the quanta of this field are identified as particles, and the force between the particles of the original matter field 1/1 is explained by the exchange of these gauge particles. This principle was first applied by Yang and Mills, and we turn to this now.
P roblems
20.2(1 ) Derive (20. 1 9) as Euler eq uations for the Dirac action .
20.2(2) Show from (20.3) that J k Vriykljr is a contravariant 4-vector field . Prove this also by looking at the transformation properties of Jk = Vr iYk ljr , using (20. 1 6) and (1 9.44) . =
20.2(3) Show that the term I Ai Ji vol 4 is gauge-invariant if J has compact support.
20.3. The Yang-Mills Nucleon
How did the groups S U (2) and S U (3) appear in particle physics?
20.3a. The Heisenberg Nucleon
Heisenberg postulated that the proton p and the neutron n behave identically with respect to the "strong" interactions between nuclei. These forces are much stronger than electromagnetic effects on the charged proton. Suppose then, with Heisenberg, we neglect completely all electromagnetic properties. He then considered p and n as being two states of the same particle, the "nucleon," represented by two 4-component spinor functions, again denoted by p and n. We shall not here be concerned with the spinor components, but shall write schematically 1/1 = (p, n) T
538
YA NG-M I L L S FIELDS
where p and n are now complex-valued functions o f space-time. Thus a nucleon that is, in the estimation of some observer, definitely a proton at a given point wou ld have n = 0 there; a neutron would have p = O. Heisenberg felt that an observer is free to make a global linear change in the components (p , n l , keeping I p l 2 + I n l2 inv ariant" for example, the nucleon could be called a proton at any given space-time point. essence, then, Heisenberg demanded that the (unknown) strong force Lagrangian for the nucleon must be invariant under the generalized gauge transformation
fu
0/
f--7
[
u o/ = U l l U21
where, since I p l 2 + In l 2 is to be unchanged, U Since
U 12 U 22
E
][] p n
V (2) is a (constant) unitary m atrix .
(p, n l and (eia p, ei a n l represent the same nucleonic mixture we may eliminate this special phase transforma tion by restricting U to have determinant I ; the symmetry group of the strong Lagrangian then consists of constant matrices U E 5 V (2) and the nucleon admits 5 V (2) as a global gauge group. (As I learned from Meinhard Mayer, Heisenberg actually thought not in terms of 5 V (2) but rather the spin "representation" of 5 0 (3) ! )
20.3b. The Yang-Mills Nucleon Yukawa, in 1 935, introduced the idea that one should explain the strong nuclear force between nucleons by assuming that the force arises from the exchange of certain parti cles, mesons, unobserved at that time, j ust as the force between electrons results from the exchange of photons. Yang and Mills in 1 954 suggested that we can arrive at ex change mesons by assuming that the correct Lagrangian for the nucleon will admit 5 V (2) as a local symmetry group, rather than the global one of Heisenberg. Weyl's principle will then require a gauge field, that is, a connection. Recall that when we studied (in Section 1 6.4e) an electron moving in the background field of a magnetic monopole, the vector potential was not globally defined and had to be defined in patches of Mri. The nuclear field, analogous to the electromagnetic field, is completely unknown. There is a good chance that any "potential" for this field 0/ field. Thus the nucleon will again only be defined in patches, and likewise for the field should be considered not as a ((: 2 function on space-time but rather as a section of a ((: 2 vector bundle, whose structure group is 5 V (2) . Of course the bundle might be trivial, but it is no more work to consider the general case. Gauge transformations are simply changes of frames in the fibers of the bundle. In this new unknown bundle the Yang-Mills covariant derivative will be locally of the form
B \jJ· = - + wJ Bx j where Wj = w (8/8xj) and w = (Wa b ) = dxj w/ b is an ML(2) -valued connection
I -form . .uL(2) consists of skew hermitian matrices with trace 0 and so has a basis consi sting of imaginary multiples of the Pauli matrices {i aa } , a = I , 2, 3
[ 0i -i ] 0
'
T H E Y A N G -M I L L S N U C L E O N
539
Thus each Wj i s o f the form
Wj
=
. - l q Ua AUj
= =
.
- l q U · Aj
-i q {u I A ; + u2 A � + u3 A � }
(20.25)
where we have completely suppressed the matrix indices. We have thus been forced to introduce three new covariant vector fields A I , A 2 , A 3 , the Yang-Mills fields, to mediate the force between nucleons. The strength of the force is reflected in the coupling constant q , replacing the charge in the case of electromag netism. Our covariant derivative is
V1· 1fr =
(J 1fr
(hl
.
- l q U · A1 1fr .
·
(20.26)
where again 1fr = (p, n) T One then must introduce "kinetic terms" in the Lagrangian involving derivatives of the A fields, that is, of the connection w. The natural candidate for "derivative" of W is of course the curvature
e
=
1 dw + 2: [w, w]
=
dw + w !\ W
Then
=
ejk
=
(Jj Wk - (Jk Wj + Wj Wk - Wk Wj aj Wk - ak Wj + [Wj , wd
(20.27)
(Caution: Each Wj is an ordinary matrix, not a matrix of I -forms ! ) Introducing the matrices we get
(20.28)
where
Fjk
aj A k - ak A j - iq [ A j , Ad is again a trace-free hermitian 2 x 2 matrix, the field strength of the Yang-Mills field. :
=
We must remark that Yang and Mills were unaware, at the time, of the notion of curvature of a vector bundle; the bracket term in (20.28) was added because they knew that some term was needed to give a nonabelian version of electromagnetism ! For an interview with Yang on the history, see [ Z ] . In our former notation
ea b
- 2:I RU bj k d X
-
j !\ dX k
ejk is the skew hermitian matrix with a fl entry R a fljk . We wish to construct a sca lar from e . The analogue of the Ricci tensor R a fl a k makes no sense (why?), and and so
so the scalar curvature analogue doesn' t exist. An obvious scalar can be constructed quadratically from the Riemann tensor, namely R a fljk R fl / k (the indices jk here have
540
Y A N G -M I L L S F I E L D S
been raised by the Minkowski metric tensor), which i s essentially the trace of the m atri x L j k 8jk8 j k . One then adds to the Lagrangian a kinetic term proportional to this trace
k
tr ( Fj k Fl ) We shall discuss this more thoroughly in the next section. After second quantization the fields A I , A 2 , A 3 yield three particles, the exc hange particles that mediate the nuclear force. This model of nuclear forces is now obsolete. The currently accepted versi on ho lds that the nucleons are not fundamental; each is made up of quarks. Each "fl avored" quark 1/1 appears in three different color states 1/1 = (R , B , G) T , analo gous to the two nucleon states (p, n l . The gauge group is then the 8-dimensional SU (3). Its Lie algebra of traceless skew-hermitian 3 x 3 matrices has a basis given by {iA b } , where A b are the hermitian Gell-Mann matrices ; see p. 652. The connection is of the fonn W.I· = -igsA" A QJ , where there are now 8 covariant vector fields A I , . . . , A8, and the "charge" g, is called the strong coupling constant. There are then 8 gauge fields, the gluons, that yield the forces between quarks.
20.3c. A Remark on Terminology We have related the connection matrices w to the gauge potentials A by
w
=
-iqA
charge. Now i t follows from the transformation rul e for a connection that if w is a connection for a bundle E then a multiple aw of w is ag ain a connection for E only if a = 1 q is called a generalized
awl
=
g - I awg + ag - I dg
Thus if w is a connection, A = (i / q)w is not a connection, and it transforms in a slightly different way
AI = g- I Ag +
(�) g- I dg
In spite of thi s, physicists almost always refer to A as the connection, and F as the curvature.
=
(i / q)8
Problem
20.3(1 ) Show that if w and WI are connections for E then their convex combination (1
is also a connection for
- a)w + awl
E for each real a.
541
C O M P A C T G R O U PS A N D Y A NG-M I L L S A C T I O N
20.4. Compact Groups and Yang-Mills Action
What if the group
is
a compact group other than SU ( N ) ?
20.4a. The Unitary Group Is Compact Theorem (20.29): The group U (n) is compact Consider U (n) as the subset of complex n 2 space satisfying uut uuT = I , that is,
PROOF :
L UijUkj J
=
°ik
In particular
L I UkJ I2 = L UkjUkJ k,j k ,J
=
L Okk k
=
n
Thus U (n) consists of points that lie on the sphere II u II = .J1i and is therefore a bounded subset of complex n 2 space, It is also clear that the limit of a sequence of unitary matrices is again unitary, and so U (n) is a closed, bounded (i,e" compact) set (see Section 1 ,2a), 0
20.4h. Averaging over a Compact Group We have seen that the left and right invariant I -forms on the affine group of the line, A ( 1 ) , do not always coincide, This is to be expected in general. Let {a j } and { T J } be bases for the left invariant and right invariant I -forms on G that coincide at e, The corresponding Haar measures
a 1 !\ . . . !\ a N and T
1
!\
,
, , !\ T N
will in general be different, as they are in A ( I ) , This cannot happen in a compact group.
Theorem (20.30): In a compact Lie group, the left and right Haar measures
coincide (the Haar measure is hi-invariant),
Let w = a l !\ . , . !\ a N be the left invariant volume form and let e be an orthonormal basis of left invariant vector fields; in particular w ( e) = I . To say that w is not right invariant is to say that for some right translate, w ( eg- I ) : = w (Rg- , *e) = c -=I- 1 . B u t then w (geg - 1 ) = c, By replacing g b y g- I i f necessary we may assume c > 1 . Thus under this adj oint action Ad(g), the orthonormal e at the identity is sent into a frame at the identity with volume c > 1 . Under Ad(g" ) , the frame e is sent into a frame with volume c" -+ 00, as n -+ 00. This means that the continuous function F : G -+ lR defined by F (g) = w ( g eg I ) is not bounded on G, But a continuous real-valued function on a compact space is bounded, a contradiction. 0 PROO F :
-
542
YANG-MILLS FIELDS
Given a compact group G with bi-invariant volume form w , the integral of a c onti n uous function f : G � e is usually written
•
l fw = l f (g)wg
where Wg is the volume form at g . (This is similar to the notation I f (x )dx. ) When co has been normalized so that the total volume of G is 1 , Ie W = 1 , then Ie f w is si mpl y the average of f on G and plays a central role in many aspects of Lie theory.
Theorem (20.31): For any continuous function f and for all g in the compact group G we have
l f (hg)Wh = l f (gh)Wh l f (h)Wh =
Consider first Ie f (hg)Wh . Right translation R g : G hg. Since W is right invariant Wh = R; Whg = (R; w h
PROOF:
�
G sends h
M
that is,
RIi* W = W Also f (hg) = f 0 R� (h) = (RZ f) (h), and so the function F defined by F(h) f (hg) is simply F = R ; f. Hence
,
=
l f (hg)wlz l Fw l (R; f) R; w r fw 1e f (h)Wh 1e R; U w) JR,e =
1\
=
1\
=
=
=
since Rg G = G . The proof for Ie f (gh)Wh is similar since W is left invariant as well. 0 In many books this proof is written as follows: The statement that W is right invariant is written
Then
r f (hg)wlz 1e f (hg)Whg 1e f (h)Wh
Je
=
=
(20.32)
replacing the dummy variable hg by the dummy h .
20.4c. Compact Matrix Groups Are Subgroups of Unitary Groups Let G be a compact group of n x n matrices. We can consider the matrices as linear transformations of en (think of them as being complex matrices). Let (, ) be any hermitian scalar product in en (e.g . , (z, w) = (2:: Zj W j ) ) . The matrices will not, in general, preserve this scalar product (i.e., the matrices will not be unitary with respect to this metric). We claim, however, that the averaged scalar product will be invariant.
543
COMPACT GROUPS AND YANG-MILLS ACTION
For given X, Y i n
( X, Y) This is of the form
(X, Y )
(gX, g Y)
=
1 (h X, h Y)w"
(20.33)
Je f (h)w" . Then, from (20.3 1 ), for g E G
1 (hgX, hg Y)Wh 1 f (hg)wh =
=
(X,
y)
Thus the compact matrix group acts by unitary transformations with respect this new scalar product. After choosing a new basis for
as de sired.
to
=
:=
metric, the matrices will be unitary in the usual sense. In this sense we may consider any given compact matrix group as a subgroup of the unitary group, (More accurately, i t is similar to such a subgroup.)
20.4d. Ad Invariant Scalar Products in the Lie Algebra of a Compact Group Let G = U (n) , the group of unitary matrices, g t : = gT = g - I . Then � = tt(n) is the space of skew hermitian matrices, X t = X T = - X .
We shall always consider Lie algebras as real vector spaces.
Define a real scalar product ( , ) in the vector space IL(n) by
( X, Y)
: = - tr X Y
=
- X ij Yj i
(20.34)
(This agrees with that used for SU(2) in ( 1 9.9) . ) In Problem 20.4( 1 ) you are asked to show that this form on �/ = II(n) is real, symmetric, and positive definite. Note that this scalar product in u�n) is invariant under the adjoint action of G = U (n) on � ; for u E U (n)
(uXu - l , u Yu - l )
=
- tr u X Y u - 1
= - tr X Y =
( X, Y )
Now let G be any compact n x n matrix group. As we have seen, G may be considered a subgroup of U (n) , and then, as we have seen in Section 1 5 . 4d , �' is a subalgebra of tt(n) . Then for X, Y in �/ we will have that (X, Y) = -tr X Y is a real scalar product in � that is invariant under the adjoint action of G on 2" ! For X, Y, Z in �
t (e t x Ye - t X , e X Z e - t x )
=
( Y, Z)
Differentiating and putting t = 0 gives, from ( 1 8.32),
([X, th at is,
y),
Z) + ( Y, [X, Z J )
ad(X) : 2' � � '
= 0
(20.35) is skew adjoint
and note that this holds for any group whose Lie algebra is endowed with a scalar product invariant under the adjoint action!
544
YANG-MILLS FIELDS
20.4e. The Yang-Mills Action
Let Jr : E ---+ Mil be a vector bundle with compact structure group G C U (N) . We are mainly concerned with the case Mil = M 4 = space-time. In the original Yang-Mills model, G = S U (2) . If w is a connection in E then w = - iq A
(20 .36 )
expresses w in terms of the "gauge field" or "potential" A and a "coupling con stant" or generalized "charge" q . Since G C U (N) , Wj = w (8j ) is skew hermitian and A j is hermitian. For curvature
e
=
I
dw + 2 [w , w]
=
-iq F
(20.3 7)
Fj k = aj A k - ak A j - iq [A j , Ad and F is the field strength. It also is hermitian. In our computations we shall use w and e ; when we are done we may convert to A and F ! Our constants might differ from those used in physics. We define the Yang-Mills (briefly, Y-M) action functional by
S [ w] : = � r -tr ( ejk e j k ) voln (20.38 ) 4 JM Note that for each j, k, ejk = (R'" fJjk ) is a skew-hermitian matrix, that is, ejk E :y , and - tr (ejk e; k ) is the scalar product in :y of these matrices. The indices in e j k have been raised by g jk , the pseudo-Riemannian metric in M il . We wish to write this action using the curvature forms, rather than matrices. The curvature forms are eu
=
1
(e '" fJ ) = R '" fJj k dx & /\ dxt
Each matrix eu is a matrix of locally defined 2-forms e'" fJ . Each of these 2-forms e a fJ has a Hodge dual (n - 2) -form *e'" fJ from the pseudo-Riemannian metric on Mn , and we know from ( 1 4.6) that
R'" fJj k R" 'I jk VO In - 2 .'e '" fJ /\* e " � We can then write the action as
S[w] = - � r e"' fJ /\ *e fJ ", 2 JM =
1
=
- � r tr e /\ *e 2 JM
2 (e , e)
where we have defined a Hilbert space scalar product (,) on W by
(e p , ¢P) : = -
L tr e /\ *¢
C
(20.3 9)
LL(N)-valued p jorms
(20.40)
This makes sense whenever e and ¢ are p- form sections of an Ad (U ( N ) ) bundle since tr [cec- 1 /\ c*¢c- 1 ] = tr [ e /\ *¢] .
545
THE YANG-MILLS EQUATION
(J)
w?
How does S depend on the connection Take a I -parameter family of connections == W eE) with "velocity" := For first variation (keeping the metric on M
ow w'(O). 1 8, 8) (08, 8 ) oS[w] ddE {S[W (E) ]}E=O = 20( (0{dW + �[w, w]}, 8) (doW + �[OW, w] + � [w, ow], 8 ) (20.41 ) S'[w] = (dow + [w, ow], 8) since w and ow are I -forms ; see ( 1 8.7). Now if WI and W2 are connections their difference A{J) is a 9'-valued I -form that transforms as Awv = cvuAWUCv& and is thus a I -form section of the Ad bundle associated to the G bundle E. Likewise, D{J) is a I -form section and of course the curvature 8 is a 2-form section of this same bundle. But then dow + [w, ow] = Vow is the covariant differential of ow , see ( 1 8.42). We then have, from (20.4 1 ), (20.42) 08 = V (ow) S'[w] = ( Vow , 8) = (ow, V*8 ) where V* is the Hilbert space adjoint to V. As usual w e demand that S'[w] 0 for all variations ow of w. This gives V*8 = 0 (Yang-Mills) (20.43) with, of course V8 0 (Bianchi)
fixed)
=
: =
=
=
=
=
the latter holding for any connection. These equations clearly generalize Maxwell's equations in the case when the current J vanishes.The coordinate expressions for these appear in Section 20.5. Problem
20.4(1 ) Show that (20.34) is real, symmetric, and positive definite.
20.5. The Yang-Mills Equation
How do the Yang-Mills equations compare with Maxwell's?
20.Sa. The Exterior Covariant Divergence
W� have seen in (20.42) that the Y-M curvature 8 satI sfy
V*8
= 0, where
=
V*
-iq F, a f-valued 2-form, must
V* is the Hilbert space adj oint of�the covariant exterior
546
Y A N G-MILLS FIELDS
differential V for the Ad bundle. We shall compute a coordin ate expres si on for this analogous to the formula ( 1 4. 1 5) for scalar-valued forms. V* satis fies
J tr
(dow
+ [ w , ow] , e ) vol
=
J tr
(20.44)
( o w , V*e ) vol
for all I -form sections ow and all 2-form sections e of the Ad bundle. Here ( , ) is the pseudo-Riemannian (pointwi se) scalar product. We can also write this as
J tr
{dow + [w, ow] }
1\
*e =
J tr
ow
1\
(20.45)
*v*e
All the forms involved take their values in the fixed vector space � and b oth d and * commute with taking traces (* only affects the manifold indices i, j, . , not the fiber indices (Y, (3 , . ) . Consider the left-hand side of (20.45). The first term is .
.
.
J tr
{dow
1\
*e } =
J
dowa f3
1\
.
*e f3 a = (do wI> fl ' e f3 1»
= (owl> f3 , d*ef3 o J =
J tr
(20 .46)
k
owd d*e } vol
assuming as usual that the boundary integral involving ow vani shes. The second term on the left-hand side of (20.45) can be computed using [W , OWL k = {w
1\
ow
+ ow 1\ w} (a j , ad
= Wj OWk - WkOWj
+ OWj Wk - OWk Wj
for then [w , OWL k e J k = 2 [ wj , owde j k (since j and k are form indices, e jk = -ekj). Then
J tr [
1\
w ow] ,
1 J tr OW] jk e J k vol = J tr [Wj , OWk ]e j k vol = J ( [Wj ,
*e =
[w ,
-
j owd , e k ) vol
where (, ) is the scalar product in �/ . From (20.35) we can write this as =
J (OWk o [Wj , e jk ] ) vol J - tr OwdWj ' e jk ]vol =
Combining thi s with (20.46) gives
J tr
owd (d*e /
-
[Wj , e jk ] } vol =
J tr
ow
1\
*v*e
But from ( 1 4. 1 5 ) (d *e / = e jk Ij ' where this covariant derivative is with respect to the pseudo-Riemannian connection on M, not the bundle connection. e is to be considered _
as a second rank tensor on M with extra indices from (}Q that are not considered in t.his covariant derivative! Finally we have the coordinate expression of the Y-M eq uatIOn v*e = 0
(20.47)
where, we emphasize, all indices are manifold indices ; Wj and e jk are matrices whose indices have been suppressed
547
THE YANG-MILLS EQUATION
We re mark, though we shall have no use for it, that the expression (20.47) can be k written as the negative of a tensorial type of divergence. The � component of (20.47) j k R jk a {3 . Thus e 1 + [W , e jk ] becomes can be obtained from (e'x jJ ) jk = j j e jk I ;. + w; e j k
_
R" jk w Y e jk w;· = e jk I ;. + w;a RY fJ jk Y ; fJ Y '" r r j k R" R = (a;. a {3 jk + r ; r R fJ + r k;r {3 j ) R a jk w Y + Wa ;. Y R Y {3 jk Y jfJ _
_
Note that we could then write (20.47) as ( '<7*e ) a fJ k v
_
-
- R" {3 j k 11;
.
-
-0
(20.48)
where we are considering R �jk as the components of a tensor of type E ® E * ® T M ® T M, and I I denotes the covariant derivative of such a tensor, using w for the bundle p art and r for the tangent bundle part.
20.Sb. The Yang-Mills Analogy with Electromagnetism If we now put w = -i q A and e = -i q F , then we have seen in (20.37)
generalizes the situation in electromagnetism, where the action is ( when no sources are present) essentially J Fj k F j k vol4 . The Y-M action is, except for a constant, S[A]
....., J tr J tr
=
Fjk F jk vol"
(20.49)
( aj A k - ak A j - i q [A j , Ad ) ( a j A k - a k A j - i q [A < A k ] )vol"
Whereas the electromagnetic action is quadratic in the fields A, the Y-M action also contains cubic and quartic terms. The Y-M equation v*e = 0 and the Bianchi equation ve = 0 are, from ( 1 8.44) and (20.47), and
(20.50)
It is instructive to compare these with Maxwell's equations in Mg with metric {- I , 1 , 1 , I } . We shall write the Y-M fields for G = S U (n) as follows. We give the usual electromagnetic names to the components of F i
=
1 , 2, 3
even though E and B are now 3-vectors with hermitian n x n matrix components. Look, for example, at Y-M for k = O. We have
548
YANG-MILLS FIELDS
that is, div E = i q (A o E - E o A )
(20.5 1)
This is the analogue of Gauss's equation . We see that even though we started ou t w i thout external sources, i q (A o E - E o A)
plays the role of a "charge density." Thus the Y-M field E and the potential A c ombine to act as a source for the Y-M field ! The nonabelian nature of the structure group S U (n), that is, [A , E] i- 0, allows this to happen ! Look again at Y-M, this time considering only a spatial index k = = 1 , 2 or 3:
F OfJ 1 o + F'xf3 l a - iq [Ao, FOf3 ] - iq [ A a , F af3 ]
that is,
aE
curl B = - - iq (Ao E - E A o ) + i q (A at
x
f3
=0
B+B
x
A)
(20. 5 2 )
replacing Ampere-Maxwell . Note that there are two extra contributions to a "cu rrent" other than the displacement current. The Y-M equations thus yield generalizations of the laws of Gauss and of Ampere Maxwell, without external sources. Similarly, in Problem 20.5 ( 1 ) you are asked to derive the analogues of the laws of Faraday and of the absence of magnetic monopoles from the Bianchi identity
aB
curl E + at = i q { (A
x
E
+E x
A) +
(A oB - BAo) }
(20.53)
and div B = i q (A B - B o A) 0
(20.54)
Note that "magnetic charge density" can exist in a nonabelian Y-M field !
20.Se. Further Remarks on the Yang-Mills Equations It is clear that if ¢> is a p-form section of any Ad ( G ) bundle, then tr ¢> is an ordinary p- form on M since tr (c v u ¢>cv� ) = tr ¢>. Note that i f G = S U (N) then for any p-form section ¢> o f the Ad (G) bundle (for example the curvature 2-form) we must have tr ¢> = O. However, i f 1/1 is another form section, then ¢> /\ 1/1 does not take its values in �' . Although tr (¢> /\ 1/1 ) is again a form on M it need not be O. Furthermore, there are times when one uses groups other than
SU(N) .
Theorem (20.55): Let ¢> b e a p-form section of a n A d (G) bundle. Then
dtr ¢>
= tr
V¢>
PROOF:
V ¢> = d ¢> + [w , ¢>] = d ¢> + w /\ ¢> - (- I ) "¢> /\ w
549
THE YANG-MILLS EQUATION
an d so
tr V¢ = tr d ¢ + ui' fJ 1\ ¢ f! 01 = tr d ¢
-
D
wOl fJ
(- l ) P ¢ fJ 01 1\
The following is clearly the analogue of ( 1 4. 1 2). Recall that we are using the Hilbert space scalar product (20.40) .
Theorem (20.56): For any form section of an Ad(G) bundle
P R O O F : Let y be a (p - I )-form section of the Ad bundle with small support. Then, from ( 1 8.46)
(V y , ¢) =
-
=
-
J tr J
( V y 1\ * ¢)
dtr ( y 1\ *¢)
=
±
-
J tr
J tr
V ( y 1\ * ¢ ) ±
( y 1\ V *¢)
J tr
( y 1\ V *¢)
Since y has small support, the first integral vanishes by Stokes's theorem. We conclude
The actual sign is given as in ( 1 4. 1 2).
Definition (20.57): A Yang-Mills field A is one that satisfies V* F
Definition (20.58): Any field strength satisfies
=
0
Fjk
= oj A k - Ok A j - iq [ A j , Ad that
*F is called self-dual
F=
- * Fis called anti-self-dual
Since any field strength satisfies the Bianchi equation V F = 0, we see that V * F = 0 if F is self- or anti-self-dual . A self- or anti-self-dualfield strength is automatically the
fie ld strength of a Yang-Mills field!
Problems
------
20.5 (1 ) Supply the details of the electromagnetic analogues (20.53) and (20.54) fo r the Bianchi equations.
550
YANG-MILLS FIELDS
20.5(2) The electromagnetic analogues can also b e derived using exte rio r fo r m s . FI'I I in the details in the following. Decompose A into temporal and spatial parts A
aA1
at
E1
=
d¢
82
=
dA 1 + A 1
-
Then the Bianchi equations V F2 dE1 +
:
a 2 t
. 1 / q[A , ¢]
-
/\ =
=
A1
=
dA 1 +
�
[A 1 . A 1 ]
d F2 + [w 1 , F 2 ] =
0 will yield
iq([A 1 , E 1 ] + [
d8 2 = iq[A 1 , 8 2 ]
For the Yang-Mills eq uation V* F ±*V* F = 0 , we put * F2 = -*82 /\ dt + *E1 for �,-valued forms *82 and *E1 ; the bold * i s the spatial Hodge operator. =
Then
yields
and d*82
=
a*E 1
--
at
+ iq( [A 1 , *8 2 ] - [
20.5(3) Let M4 be compact and suppose that the support of 8 w does not meet the bou ndary (if any) of M. Use 8e = V (8w) and Theorem (20.56) to show that 8
1M tr
(e
/\
e)
=
±8 (e, *e)
=
0
Thus if JM tr (e /\ e ) is added to a given action integral , the action will be altered but the variational equations will be unchanged! We shall study the Horm tr (e /\ e) extensively in our remaining chapters.
20.6. Yang-Mills Instantons
How can the Brouwer degree distinguish between two Yang-Mills vacua?
20.6a. Instantons Consider a quantum particle interacting with a Yang-Mills field in Minkowski space. This particle is described by a "wave funtion" 1jr , a cross section of a complex rf
551
YANG-MILLS I NSTANTONS
vector bundle £ over Minkowski space M = Mri . We assume that the structural group i s S U (n) ; thus G = S U (n) acts on CN via some representation. For our purposes it is sufficient to consider the standard representation on C/ . The bundle has a Y-M connection w = - i q A and a curvature B = -i q F, where A and F are hermitian matrix valued local forms on Mri . In U C M we have a frame of sections
\lI u
=
eu
=
(ef , . . . , e� )
and Wu and Bu . I n an overlap ev = eu cu v , cu v (x ) E SU ( n ) . In this section w e shall be concerned with the background Y-M field, rather than with the particle. The action for this Y-M field alone is essentially
i -tr Fjk Fjk * 1 i '"'-'
( II
E
11 2 - II
B
11 2 ) * I
where we have given the electromagnetic analogue on the right (Problem 7.2(3)) . For certain purposes it is useful in physics to replace the Minkowski metric of space time by the 4-dimensional euclidean metric + d t 2 + dx · dx. This will not be discussed here. (See e.g., [C, chap. 7] . This chapter of Coleman ' s book will also overlap with some of the topological material that we shall discuss later.) The action is then called the euclidean action. We shall be concerned with Y-M fields having finite euclidean action
(Note that the euclidean version of the electromagnetic Lagrangian is the energy density
of the electromagnetic field.) Such fields are called instantons since they "vanish" as I t I -+ 00. An example of an instanton is given in [I, Z, sec. 1 2- 1 -3].
For simplicity, to avoid the limiting values of boundary integrals, we assume that the field strength £ 2 + B 2 not only dies off at infinity but has support lying inside some 3- sphere S 3 centered at the origin of ]R4 .
Figure 20.1
(This does not make sense in electromagnetism in Mri since an electromagnetic field
in free space would radiate out to infinity and would be present for all t .)
552
YA NG-MILLS FIELDS
Let U b e a coordinate patch holding this S 3 and its interior and let V b e a Coordinate patch holding S 3 , extending to 00, and such that F = 0 in V . We assume that V is the exterior to some sphere inside S 3 . In the "exterior region" V we have = O. We claim that we can make a c h ange of frame over all of V (in the wave "function" vector bundle E, not in Min kow ski space) so that in the new frame = O! This should not b e a complete surprise; i t i s a global version of Riemann ' s the orem be the original (9.70) on curvature 0, but for an arbitrary vector bundle. To see it, let connection form for V . We wish to find a : V --+ S U (n) so that
e
Wv
Wi
g
that is,
dg + w'g Can we solve this I -form system for g g;(x)? Using the symbol to (dg wi g) as arises in the Frobenius theorem d (dg + wig) dw'g - Wi dg dw'g - Wi (-wig) (dw' + Wi w') g e'g in V =0
=
+
=
�
(20.5 9) signify mod
A
A
�
= 0,
�
A
�
By Frobenius we may locally solve (20.59) uniquely for g , subject to any initial go
g (p) at p E V . dg
g and h, i n two overlapping patches. -wig dh -w'h , d(g-lh) _g-l dgg -1 h + g- l dh g - lw'gg-1h - g-]w'h
Suppose that w e have two solutions, = and = and so
=
Then
=
=0
=
Thus two overlapping solutions are always related by a constant matrix k E SU (n), h = gk , at least if the overlap is connected ! Consider then a path C : [0, I ] --+ V that starts at p . Cover this path by a finite number of 4-balls (lying in V ) each small enough to support a solution to (20.59) and such that the intersections of consecutive balls are connected. Let be the solution in the first ball at p. Let be a solution in the next ball intersects in a connected set. Then there is a constant matrix E SU (n) such that = in their overlap and it is clear that g ; : = is a new solution of (20.59) in B I that agrees with in their overlap. We have continued the solution into the second ball. Proceed to the third ball and so forth. In this way we continue the given solution in the initial ball to all points of V . Is this well defined? If C is a closed curve that returns to p, the final solution could be a that differs from this is the same situation as in analytic continuation of an analytic function in the
k]
go;
gO! go B ] . B] Bo g] (x) go(x)k]
BO!
Bo
g]
g]k1 ]
go
gb
complex plane ! However, the region in �4 that is exterior to a ball i s simply con ne cted, and just as analytic continuation i s unique in such a region (seen by shrinking the closed
YANG-MILLS INSTANTONS
553
curve to a point), so it is in our situation. Thus a global solution g : V � SU (n) to (20.60) exists in all of the exterior region V and Wv
=0
when we use the new frame of sections e v = e/v g. 0 Note that the original connection WI is of the form WI =
_ dgg - 1
(20.60)
and is said to be pure gauge. Since Wv = 0 on U n V, and in particular on S 3 ,
We again write this in a simplified form, C v u =
g, (20.6 1 )
where g : S 3 � S U (n) are new matrices, not those of (20.60). We then have the following situation: Look at the part of the wave function bundle that lies over the sphere S 3 . Over S 3 we have two frame fields given, the "flat" frame v ev and the frame e u over U . The flat frame consists of sections e I , . . . , e,; each of which is covariant constant
that is, these sections are parallel displaced along S 3 . We are comparing the U -frames eu with these covariant constant frames along S 3 , e u (x ) = e v (x ) c v u (x )
= ev (x ) g (x )
(20.62)
and, consequently the matrices g(x) define a mapping
g : S3
�
SU ( n )
(20.63)
This situation is similar to that encountered in Chern's proof of Poincare 's index theorem (17.2 1). Let us go back and reconsider Chern 's proof in the light of our Y-M field with fi nite action.
20.6b. Chern's Proof Revisited Consider, instead of a closed M 2 as in Section 1 7 .3, a curved "wormhole" version M 2 of the plane, but such that the curvature vanishes in the region V exterior to some circle S I . The bundle we are considering is the tangent bundle T M 2 to the orientable surface
554
YANG-MILLS FIELDS
Figure 20.2
M 2 , but considered as a complex C line bundle. By using "orthonormal" frames eu , ey,
we may consider the structural group of this bundle to be U ( 1 ) . We have indicated the "flat" covariant constant frame ey in the exterior region. Warning: Unlike the case when M il has dimension n :::: 3, the region V is not simply connected. One cannot always find a global flat frame in this region V . For example, M 2 is flat in the conical region in the following figure, but a parallel displaced vector will not return to itself after traversing S l
Figure 20.3
as we saw in Section 8.7. In fact this picture is the geometric analogue of the Aharonov Bohm effect, discussed in Section 1 6.4f. Using the electromagnetic connection, the curvature inside the coil is constant, since the magnetic field B is constant there; this corresponds to the constantly curved spherical cap. Furthermore, the exterior to the coil corresponds to the flat conical region. S ince w = -i ebde /2nn in the exterior region, the equation of parallel translation in the electron wave function bundle is d1/f - i eb1/fde /2nn = O. Hence 1/f = exp(i ebe /2nn) is covariant constant but is not single-valued unless the flux b takes on very special values !
555
YANG-MILLS INSTANTONS
(Notational comment: In the case of a section of a vector bundle with structure group G , parallel translation along a parameterized curve x = x (t ) is still defined by d1f; + w1/! = 0, that is, d1/! dX (20.64) = _w 1/! dt dt for the matrix-valued connection I -form w. Since J w (dx /dt)dt also lies in g, we see from Problem 1 5 .2(3) that if the structure group G is commutative then the solution to (20.64) is 1/! (t) = exp[ - J� w (dx/dr)dr ] 1/! (O) . If G is not commutative, there is no such formula, but physicists write the solution in the form
( )
The symbol P indicates an operation called path ordering. It is important to realize th at this can simply be considered a notation for the operation that sends an initial 1f; (0) into the unique solution 1f; (t) of (20.64). (We shall not use this notation.) In our wormhole, Figure 20.2, we have chosen V so that a global covariant constant frame ev does exist, as it does in the Y-M example. In the curved region U we have indicated a cross section eu that has singularities at the critical points of the height function; the top p is one of them. (The field looks like the normalized velocity field for molasses oozing down from the top.) For our complex line bundle version of the tangent bundle we have, as in 1 7.3a, the connection w and curvature
(20.65)
B = dw = -i KdA On the circle S I we have eu = ev e i a and so g (x ) = e ia
(20.66)
and
In the situation of Poincare's theorem, Chern considered a closed surface. In our case
1 KdA fu KdA =
since K vanishes outside U . In Chern's proof
1 KdA M
=
2 JT L ip (eu) p
whereas in our nonclosed M 2 , using KdA = dW 12 , Chern's proof would give
1 1 KdA
2JT
U
=
""" .
I i Wul 2
0 i p (eu) + 2JT p
= L ip (eu) p
- _2JT1 ls,1 da s'
(20.67)
556
YANG-MILLS FIELDS
We may then write (for future reference)
� fL e = L jp (eu) + 2� is, w
2 or
p
Jr{ e
i = _ 2]'( }M
L ip (eu)
_
p
I 1 da 2 ]'( 's'
(20.68)
_
(20.68) tells us that we get the same result as in the closed M 2 case except for a boundary term describing how many times the given cross section rotates around the flat section ev !
_
1 1
_
2 ]'( ls,
(20 .69) d L (e v , e u )
Note that this last "rotation number" is exactly the degree of the map
g : Sl
�
S l defined by x
� g (x ) =
ei a
Now in our Y-M situation we have a similar map, at least in the case when G
S U (2) , for then (20.63) involves a map g : S3 � SU (2) = S3
=
(20.70) and this map indeed does have a degree, called the winding number of the instanton. In our Y-M case we shall assume that the frame e u in the wave function bundle has no singularities inside S 3 . We draw a surface analogue consisting of a flat cylinder V with a hemispherical cap (a diffeomorphic copy of �2 ) U . In V we put the flat vertically oriented "frame" ev, whereas in the cap U we may put a singularity-free field eu, for example, as follows. In Section 1 6.2a we introduced a vector field on S 2 having a single singularity of index 2 at the north pole. The field eu is simply the part of this field that lives on the southern hemisphere.
Figure 20.4
557
YANG-MILLS I NSTANTONS
Problem 20.6( 1 ) you are asked to verify (20.69) in this case. In th e Chern situation, in (20.69), if there are no e u singularities, we see that the degree of the boundary map is completely described by the integral of the curvature ! In
Can the corresponding Y-M degree in (20. 70) be evaluated by looking at the curvature, that is, the field strength, of the Yang-Mills field? The answer yes will be proven in
Sec ti on 2 1 .2; it was first given by Chern a decade before the paper of Yang and Mills.
20.6c. Instantons and the Vacuum
In Yan g-Mills we may consider the
vacuum state in which the field strength F or e
vanish es. One must not conclude that nothing of interest can be associated to such a
v acuum. In the geometric analogue we may consider a flat surface ; the connection (i) replaces the gauge field A and the curvature e = 0 replaces field strength F = O. In the example considered previously of the frustrum of a flat cone, tangent to a 2-sphere alo ng a small circle S I , we may delete the spherical cap completely. This corresponds to the exterior region in the Aharonov-Bohm effect. We have seen that parallel translation I about S does not return a vector to itself, in spite of the fact that the connection is flat. There is more information in the flat connection than is read off from the 0 curvature alo ne ! Likewise there is more information in a gauge field A for a vacuum than can be read from the vanishing field strength. Before considering the Yang-Mills vacuum we shall look at another geometric analogue. In the following figure we have again drawn the 2-dimensional analogue, a flat surface, but instead of using the "flat" (covariant constant) frame (pointing, for example, constantly in the t direction) we use a frame that is time-independent, is flat at spatial infinity, and rotates (in this case) once about the flat frame along each spatial slice .
./\ .1\ /-_.
V
.�
·
·
I.. .:.....
....L...
·
.f-
----.. ----..
flat frame
V
./�
..,�
space
L
"me
e = �
Figure 20.5
We have gauge transformed the flat frame e to a new one, eg, where g : JR -+ S I maps each spatial slice so that g ( -00) = g (00) = 1 . (The field, i.e., connection, is again "pure gauge," (i) = g- l dg.) We assume, again for simplicity, that for each spatial section t = constant we have g (x ) = 1 for Ix l 2: a for some a . This vacuum solution in JR2 is not deformable, while remaining flat at spatial infinity, into the identically fl at frame vacuum for the following reason. U(1) =
558
YANG-MILLS FIELDS
The function g maps the spatial slice lR into S l . We may stereographically proj ect lR onto a circle S l -(north pole) by projecting from the north pole. In this way We may consider g as being defined on S I -(north pole). Since g is identically 1 in some neighborhood of the pole we can extend g to the entire circle S I . (This can be tho ught of in the following way. By identifying all x for I x I 2: a with the point x = a on the section t = constant, this section becomes topologically a circle S I . We have "compacti fied" lR to a circle and since g = 1 for I x I 2: a, g extends to this compactification .) This gives, for each t , a map g : S I -+ U ( 1 ) = S I , which in this case has deg ree 1 by construction. If our vacuum solution were to be deformable to the flat vacuum so lution while keeping I x I 2: a flat, then 1 = deg g : S I -+ U ( 1 ) = S I would have to eq that of the flat vacuum case, which clearly has degree O. This i s a contradiction . We thus have two inequivalent vacua . Similarly, we could get a vacuum frame th at winds k times around the flat frame. In the 4-dimensional Yang-Mills case (with G = S U ( 2 » there will likew ise be an infinity of inequivalent vacua, each one characterized by the degree or "windin g num ber" of the map g : S 3 -+ S U ( 2 ) = S 3 arising from the spatial slice ]R3 "compactified" to S 3 ; this is discussed more in Problem 20.6( 2 ). Physicists then interpret an instanton with winding number k, that is, degree k given in (20.70), as representing a nonvacuum field tunneling between a vacuum at t = - 00 with winding number n , and a vacuum at t = + 00 with winding number n + k (see [C, L, sect. 1 6.2] or [I, Z sect. 12- 1 -3]). We discuss the geometry of this situation in Problem 20.6( 2 ). Further significance of the winding number of the instanton will be sketched in Chapter 2 1 . We have seen why g : S 3 -+ S U ( 2 ) has a degree. To understand why g : S3 � S U (n ) , n 2: 2 , has an associated "degree," and to understand Chern' s results when there are singularities, we need to delve more into topology, in particular the topology of Lie groups, "homotopy groups," and "characteristic classes." Homotopy groups arise also in other aspects of physics (see, e.g., [MiD. We shall proceed with this program in the next chapter.
uai
P roblems
20.6(1 ) Verify (20. 69) in the case of our specific example of the cylinder with a cap.
20.6(2) Consider an instanton. Let eu be the frame in the i nterior U; we shall assume that eu can be extended to be a nonsingular frame in all of ]R.4 . Let ev be a flat vacuum frame in the exterior V of the instanton, and let, as in (20.70), 9 : 53 ---+ 5 U (2) , mapping the su rface of the instanton i nto the group, have degree k. Recall that k is called in physics the winding number of the instanton. (i) Show that if e v can be extended to a frame on all of ]R.4 then k 0 (H int: Generalize Problem 8.3(9) . ) . Thus i n general e v cannot be extend ed. Consider a 3-dimensional "can" W3 su rrounding the instanto n, l yin g en tirely in the vacuum region V, and with ends D and D* at two spatial sli ces t ±" 00 " . Let the side of the can be given by II x 11 = a. ==
=
YANG-MILLS INSTANTONS
559
instanton with boundary 53
Figure 20.6
9 is defined on the can W3 and, in fact, on the enti re 4-dimensional region that is inside the can and outside S3 . Assume that g takes a constant value,
say 9 e, on an entire region I x II ::: (a - E) containing the sides of the can. The can then can be smoothed off near the ends D and [)* , yielding a =
smooth 3-dimensional manifold diffeomorphic to a 3-sphere and such that e everywhere on this new can except on the portions of D and [)* where g =
I I x ll < a - E .
We shall now apply the theory of the Brouwer degree. 9 maps the 3-disc S U (2) = S3 and maps a D i nto a single point 9 = e . This means that if, in D, we identify all of a D to a single point (the "point at 00" ) then we can consider this new space as a 3-sphere, and we have a map 9 of this 3-sphere into SU(2) . This map has a Brouwer degree that can be evaluated by looki ng at i nverse images of some regular value u E SU(2 ) , u =P e . Call this degree deg ( -00) n. Simi larly we can look at the disc [)* and assign a degree deg (+oo) n + k, for some integer k. I n physics books these integers are called the winding numbers of the vacua at t = - 00 and at t = +00, respectively. On the other hand, the entire can W3 is a smooth version of a 3-sphere, and we have the degree of 9 mapping this can into SU(2) . The 2-dimensional analogue is
D into
=
=
Figure 20.7
560
YANG-MILLS FIELDS
(ii) Show why
W3
deg ( g :
�
SU(2»
=
deg (+oo) - deg (-oo)
(iii) Show why this degree k is also the winding number of the insta nto n. 20.6(3) The Winding Number of a Vacuum. Let A SU(2) map a spatial section IR3 of IR4 , by
g(x)
= exp
i- 0 be any constant an d 9 : 1R3 �
[( I x i:2X;�) 1/2 ]
(x)dg(x),
We can think of this 9 as defining a gauge transformation of the c l assical g- 1 in the s patial vacuum (where w = 0) to a new one with w section IR 3 defined by f + 00 . We claim that this vacuum has windin g nu mber = ± 1 . To show this we first show that tends to a constant SU (2) group element limit (i ndependent of x) as 00 .
I x Ig(�X)
=
=
(i) What is this limit? (H int: Use ( 1 9.20) , which holds for unit A.) Now we are allowed to compute the winding nu mber using (8. 1 8) .
x
x
(ii) Show that only the origin 0 i s mapped by 9 onto I E SU(2) and show that fA, where A i s a o is a regular point by using (1 9.20) applied to the line unit vector. We have then shown that this vacuum has winding num ber ± 1 . =
=
In [I ,Z] , sect. 1 2- 1 -3, an instanton solution that tunnels between with winding n umber 0 and the vacuum of this problem is given.
a
vacuu m
CHAPTER
21
Betti Numbers and Cove ring Spaces
2 1 . 1 . Bi-invariant Forms on Compact Groups
Why is it that the I -parameter subgroups of a compact Lie group are geodesics?
Samelson's article [Sam] is a beautiful exposition on the topology of Lie groups as it was known up to 1 95 1 .
21.1a. Bi-invariant p-Forms Recall that a form or vector field on G is said to be bi-invariant if it is both left and right invariant. For example, on the affine group G = A ( 1 ) of the line, dx / x is bi -invariant.
Theorem (21.1): If a,P is a hi-invariant pjorm, then a is closed, da
=
0
Let () ' , . . . , (} n and r ' , . . . , rn be bases of the left and the right in variant I -forms, respectively, and let (} j = r j at the identity. Since the left and right structure constants are negatives of each other (see Section 1 5 .4c), d(} i = - 1 /2Cj k (} j 1\ (} k and d r i = 1 /2Cj k r j 1\ r k . Let aP be bi-invariant PR O O F :
where al are constants. Since a is also right invariant,
Now compute da at e from both expressions.
561
D
562
BETTI NUMBERS AND COVERING SPACES
21.1b. The Cartan p-Forms In Section I S. 1 a we have defined the Maurer-Cartan matrix of 1 - forms
Q : = g- l dg
[ ; �] [dX dY ] [dol !!.l.] [ dX�dY ]
When G is the affine group of the line, G = A ( 1 ) ,
Q=
We can also consider exterior powers Q 2 = Q x
x
o
0
/\
x
x
0
0
/\
Q , Q3 =
�
0 0
. .
. . For example,
x
0
which has the left invariant volume form for its only nontrivial entry. We define the Cartan p-forms Q l , Q 2 , " " Q Il =dim G by
Q p : = trQ P = tr {g - l dg
/\
g- l dg /\ . . . /\ g- l dg }
(2 1 .2)
These are, of course, (scalar) left invariant p-forms on G . For G = A ( 1 ) , Q 1 = dx/x and Q 2 = O.
Theorem (21.3): The Cartan p-forms are hi-invariant, and hence closed, dQ p = O. Furthermore, Q 2 p = O. PROOF:
For constant k E G,
l
/\
(gk) - l d (gk) /\ . . . } = tr{k- 1 (g - l dg /\ g - l dg /\ . . . g - l dg)k} tr { (gk) - d (gk)
= Qp
and so they are also right invariant. Next note Q 2 = tr(Q /\ Q) = Q ij /\ Qj i - Q j i /\ Q ij = - Q 2 , and so Q2 = O. Similarly, Q 2 p = 0, all p. D
=
The Cartan 3-form plays an especially important role. S ince Q (X) = X, all X
E
(Q /\ Q) (X, Y) = Q (X) Q (Y) - Q (Y ) Q ( X) = [X, Y ] and thus
(Q
/\
Q) /\ Q (X, Y, Z) = [X, Y]Z + [Z, X]Y + [ Y , Z]X
9"
( 2 1 .4)
Taking the trace of this and using [X, Y]Z = XYZ - YXZ , and so on, give
Q 3 (X, Y , Z)
=
3tr( [X, Y ]Z)
(2 1 .5)
When G is compact we can express this in terms of the Ad invariant scalar product (20.34) in �/ C Ll(N )
Q 3 (X , Y , Z) = -3 ( [X, V] , Z)
(2 1 .6)
BI-INVARIANT FORMS ON COMPACT GROUPS
(21 .4) brings up a point. Consider
563
G = S O (3), and let {E d be the basis ( 1 9. 1 ). Then (2 1 .7)
and this matrix is not in the Lie algebra ��(3) ! (Recall that in Section 1 8. 1 b we defined
the
bracket of 3'-valued forms to remedy this situation.) The matrix (2 1 .7) is called a
Casimir element.
21.1c. Bi-invariant Riemannian Metrics be a scalar product in 3' that is A d invariant; for example, when G = U (n), -trX Y . Thus the Lie algebra o f every compact group has such a n invariant product. Define then a Riemannian metric on the group G by "left translation," sc al ar that is, Let ( , ) e
(X, Y ) e
=
By construction, this metric is left invariant. We claim that it is also right invariant. For
by Ad invariance ! We have shown
Theorem (21.8): There is a bi-invariant Riemannian metric on every compact
Lie group. =
dx /x and a 2 = dy / x are left invariant. Hence dx 2 + dy 2 a l 0 a l + a 2 0 a 2 = ---x2
The group A ( l ) is not compact. a I
is a left invariant Riemannian metric on A ( l ) . (Note that this is the Poincare metric on the "right half plane"; see Problem 8.7( 1 ). ) This metric is not right invariant, and in
fact there are no bi-invariant metrics on this group.
Theorem (21.9): In any bi-invariant metric on a group, the geodesics are the
l-parameter subgroups and their translates.
PROOF :
Let X be a left invariant field on G . We shall show that each integral curve of X is a geodesic in a bi-invariant metric. Since X generates right translations, X i s a Killing field (see Section 20. 1 c) . Let C b e a geodesic that is tangent t o X at a point g. We need only show that X is everywhere tangent to C . By Noether's theorem, X and the unit tangent T to C have a constant scalar product (X, T) along C . T has unit length and X, being left invariant, also has constant length. Since X and T are tangent at g, it must be that X and T are tangent everywhere along C. 0
564
BETTI NUMBERS AND COVERING SPACES
Thus i n a group with a bi-invariant Riemannian metric, a geodesic throu g h e i s of the form exp (tX) , where X is the tangent at e. This was the (very meager) motivatio 11 for denoting geodesics in a Riemannian manifold by exp(tX) ! One says that a Riemannian manifold Mil is geodesically complete if every geodesic segment e (t) = exP C ( O) (tX) can be extended for all parameter values t . The euclide an plane ]R 2 is complete but the euclidean plane ]R2 0 with the origin delete d is no t; the geodesic exp ( - 1 . 0) (t a/ ax ) does not exist for t = 1 because of the hole at the origin . The Poincare upper half plane is complete; even though there is an edge at y == 0, this edge is "at an infinite distance" from any point of the manifold. It is a fact that if M is compact then it is automatically geodesically complete. Furthermore -
Theorem of Hopf-Rinow (21.10): If Mil is geodesically complete, then any pair ofpoints can be joined by a geodesic of minimal length. For a proof of these two facts see Milnor's book [M] . In a compact group G we may introduce a bi-invariant metric, and then the 1parameter subgroups are geodesics. Thus
Theorem (21.11): Every point in a compact connected Lie group G lies on at least one I -parameter subgroup. As we have seen in the case G = 51 (2,
]R) in Problem 1 5 . 3 (2) , compactness is essential.
21.1d. Harmonic Forms in the Bi-invariant Metric
Theorem (21.12): In a bi-invariant metric on a compact connected Lie group G, the bi-invariant forms coincide with the harmonic forms. The proof will be broken into several parts.
Lemma: In a bi-invariant metric, the Hodge * operator commutes with left and right translations * 0 R*g = R*g 0 * We wish to show that L ; *{3gh = *L;{3gh for every form {3 at every point gh. Thus it suffices to show that for any form a at h we have alz /\ L ;*{3gh = alz /\ *L ; {3g lz . Define aglz by ah = L ;agh ' S ince the metric is bi-invariant, so is the volume form w. Recall that (a /\ *{3 ) glz = ( agh , {3glz ) Wg h . Then PROOF:
alz /\ L ; *{3gh =
L; ag /\ L ; *{3glz h
=
= L ; ( ( a gh , {3glz ) Wg lz )
L; �a /\ *{3g h ) gh
BI-INVARIANT FORMS ON COMPACT GROUPS
565
= ( L ;(X g h , L ; {3g h )Wh = ( (Xh , L ; {3gh )Wh = (Xh /\ *L ; {3g h as desired. Similarly for right translations.
0
Lemma: Bi-invariant forms are harmonic in the bi-invariant metric. PROO F :
If {3 is bi-invariant then {3 is closed, d{3 = 0. From our previous lemma, *{3 is also bi-invariant; for example, L ; *{3g h = *{3g shows that *{3 is left invariant. Then d*{3 = 0, showing that {3 is harmonic. 0
(2 1 . 1 2) will then be proved when we show
Lemma: Harmonic forms in the bi-invariant metric are bi-invariant if G is con nected. P RO O F :
First note that a left (right) translate of a harmonic form is harmonic, since d (L ; h) = L ; dh = ° and d(*L ; h ) = dL ; *h = L ; d*h = 0, because *h is also harmonic. We claim that if G is connected then in fact L ; h g k = h k . and so on. To see this, we need only show that both h and L ; h have the same periods; see Corollary ( 1 4.27). Let z be a cycle on G and let g et) be a curve in G joining e = g (O) with g = g ( l ) . Then
But (g(t)z } , for 0 :::: t :::: I defines a deformation of z = g(O)z into gz = g ( 1 ) z ; thus these cycles are homologous, gz - z = a c , b y the deformation theorem ( 1 3.21), and since h is closed
as desired.
0
21.1e. Weyl and Cartan on the Betti Numbers of G The center of a group G is the subgroup of elements that commute with all elements of the group. For example, the center of U (n ) is the I -parameter subgroup e i IJ I , whereas the center of SU (n) consists of the n scalar matrices A I , where A is an nth root of unity.
Weyl's Theorem (21.13): Let G be a compact connected group. Then the first
Betti number vanishes, b l (G) I-parameter subgroup.
=
0, iff the center of G does not contain any
(In particular, b l = 0 for SU (n) but not for U (n ) . )
566
BETTI NUMBERS AND COVERING SPACES
P R O O F : Suppose first that the center of G contains a I -parameter grou p e'X where XE � . Then e' x g = get X for all g in G . Differentiate with respect to t and pu t = 0, yielding Xg � g . Then the left i �v�rian� vector field Xg = Lg.X == gX . . on G IS also nght Invanant, and thus bl-Invanant. In terms of a bi- invariant Riemannian metric on G, the covariant version of X, that is, the I -form (l defi ned by (l ( y) = (X, V), is bi-invariant and hence harmonic. By Hodg e's theorem
;
�
bl
2:
1.
Suppose b l #- O. In a bi-invariant metric, there is then a harmoni c, hence bi invariant I -form (l #- o. Its contravariant version is then a bi-invariant ve ctor fi eld X, that is, gXe = Xeg. Thus for all real t, gtXeg- 1 = tXe . Then exp(t Xe) == exp (gtXe g - l ) = g ex p (tXe)g - l . Thus ex p (tXe) is in the center of G . 0 Since the center of 5 0 (3) consists only of the identity, Weyl's theorem yie lds b == 0 l for G = 5 0 (3) . Of course we knew this from ( 1 3 .25) and the fact that S O (3 ) is topologically �p3 . Although the first Betti number vanishes, 5 0 (3) is not simply connected. We shall see in Section 2 1 .4 that a strengthening of this vers ion of Weyl's theorem will yield information about the contractibility of closed curves in groups. The following plays an important role in gauge theories, as we shall see i n Sec tion 22. 1 .
Cartan's Theorem (21.14): If G is a compact nonabelian Lie group, then the Cartan 3-form is a nontrivial harmonic form. In particular b 3 (G)
#-
O.
Y , Z) = - 3 ( [X, V] , Z) . We need only show that it is not identically O. But the only way ([X, V] , Z) can be 0 for all Z is if X Y - YX = [X, Y] = 0 for all X and Y in �f . B ut then, since X and Y commute, the power series shows e X e Y = e x+ y = e Y e X PROOF:
Q 3 is bi-invariant, hence harmonic, and Q3 (X,
In a compact connected group each g E G i s an exponential, and so G is abelian. 0
Finally note the component form o f Q3 . Let e be any left invariant basis and let a be the dual basis. In Problem 2 1 . 1 ( 1 ) you are asked to show that (Q3) ij k
= -3Ck ij � - 3Cijk
and thus I
Q "o' = - - CI"J"k O'
2
i
/\ 0' J" /\ 0' k
where Clj are the structure constants and where Ck ij : = gkI C;j . When we use the bi
invariant metric tensor to lower the top index of the structure constant symbol, the
T H E F U N D A MENTAL G ROUP A N D C O V E R I N G SPACES
567
coefficients Ckij are skew symmetric in all indices, not just i and j ! This need not hold when G is not compact. ul g res tin
P roblem
-
2 1 .1 ( 1 ) Com pute the preceding component form of [2 3 .
2 1 .2. The Fundamental Group and Covering Spaces
In what sense does the torus cover the Klein bottle?
21.2a. Poincare's Fundamental Group
Jr} (M)
Let Y be a closed curve on a connected space M that begins and ends at a given base point Po . Such a curve can either be considered as a map of a circle into M (that passes through Po) or as a map Y : [0, 1 ] ---+ M with Y (0) = Po = Y ( 1 ) . The latter seems more conveni ent. Consider now all such maps with the same base point. We shall identify two suc h "loops" YI = YI (e ) and Y2 = Y2 (e ) , saying they are homotopic,
provided they are homotopic via a homotopy that preserves the base point; thus there is an F : [0, 1 ] x [0, 1 ] ---+ M, F = F (e , t), with F (O, t) = Po = F ( 1 , t) for all o � t ::: 1 , and F(e, 0) = Y I (e ) , F (e , 1 ) = Y2 (e) . t is the deformation parameter. We talked about this notion in Section 1 0.2d. If Y is homotopic to a constant, we say Y is trivial and write g '" 1 .
y Po
o Po
Figure 2 1 . 1 Note that in the left-hand figure, the loop Y is not trivial as far as homotopy is concerned (try to contract it to the point Po ! ) even though it is trivial in homology (it is the boundary of an orientable surface).
568
BETTI NUMBERS AND COVERING SPACES
Given two loops YI and Y2 o n M, b y reparameterization (so that each loop is tray with double speed) we may compose them to give a new loop, which is traditio
� na
written from left to right
1 < fJ < for 0 - 2 1 : = Y2 ( 2 fJ - 1 ) for - < fJ < 1 2 - One can show that if y{ YI and if Y; Y2 , then y{ y; classes of loops on M form a group under "multiplication" �
�
( Y " Y2 )
---'?
�
ly
YI Y2 . The homotopy
Y' Y2
This is the fundamental group of M, written Jl'1 (M; Po ) . It turns out that i n a certain sense the resulting group is in fact independent of the base point, and one si mply writes
Jl' 1 (M) The identity 1 in this group is the homotopy class of the trivial loop (contractible to Po). The inverse to a loop Y is the same loop traversed in the oppo site direction, y - l ( fJ ) : = y ( 1 - fJ ) . A space i s simply connected i f all loops are contractible t o a point, that is, if the group Jl' , (M) consists only of the identity. Consider loops on the circle M ' = S' , and the resulting Jl' 1 (S ' ) . These are homotopy classes of maps Y : S ' ---'? S ' . We know that homotopic maps of the circle into itself have the same (Brouwer) degree; see Corollary (8. 1 9) . It can also be shown, though it is more difficult, that maps of Sl into itself having the same degree are homotopic. Thus a loop Y is characterized, as far as homotopy is concerned, by its degree ( i . e . an integer). Since the map fJ ---'? nfJ has degree n, we have ,
Jl', (S ' )
=Z
It can be shown that the fundamental group of the 2-torus is generated by the familiar
A and B of Figure 2 1 .2. Briefly, any loop in the rectangle can be deformed (pushed) out to the edge. Jl'1 (T 2 ) is abelian because it is clear that the loop A followed by B A
B
B
B
K
B
Figure 21 .2
followed by A - , followed by
B- ' , being a loop going around the edge of the rectangle,
569
THE FUNDAMENTAL GROUP AND COVERING SPACES
I is c ontractible to Po , that is, homotopic to the constant map; A B A - B - 1 = 1 or 2 /l B == B A . Thus Jr l (T ) is the abelian group with generators A and B . For the Klein bottle K , o n the other hand, we have, from Figure 2 1 .2, A B = B - 1 A . e W say th at Jr l (K) is the (nonabelian) group with 2 generators and the single relation /l BA - I B = 1 . The rotation group in the plane, S O (2) , is topologically S I . Jr l ( S O (2) } = Z. The
rotati on group in space, S O (3), is topologically lRp 3
Figure 21 .3
The I -parameter subgroup A of rotations about the z-axis is not contractible, A -=I- 1 , but A 2 = A A = 1 ; see Section I 9.2a. Thus (2 1 . 1 5) As we also have seen in Section 1 9.2a, that is why spinors can exist !
21.2b. The Concept of a Covering Space We have
discussed the notion of covering space informally several times in thi s book; now w e shall need to be a bit more systematic. We shall say that a connected space M is a covering of the connected M, with covering or projection map Jr : M ---f M, if each x E M has a neighborhood V such that the I preimage Jr - ( V ) consists of disjoint open subsets ( Va l of M, each diffeomorphic, under Jr : Va ---f V, with V .
We illustrate this i n the case M = lR ; M = S I i s the unit circle i n the complex plane, and Jr is the map Jr (x ) = exp(2Jr i x ) -2 ,
( V_ 2 )
-1 •
( V_ I+ l -...·--+ ( V o -tl---+( VI l
0
Figure 21 .4
2 •
( V2 )
3 •
570
BETTI NUMBERS AND COVERING SPACES
We have indicated a neighborhood U o f - 1 E S I and the preimages o f U in R The notion of covering space can also be described in terms of fiber bundle s follows:
as
A covering space of a manifold M is a connected space M that is a fiber bundl e over M with fiber F a discrete set of points .
If F has k points we say that M is a k-fold or k-sheeted cover of M . Thus � is an infinite fold cover of S I . The "fiber over I E S I " is the infinite set of integers in R The edge of a (finite) Mobius band is a circle M = S that is a 2-fold cover of the central circle M of the band
- I/ � : "F "" ��------.;� / .
M=S
M=S
I
__________
F i g u re 21 .5
The n-sphere S" is a 2-fold cover of the projective pll (JR.) . S U (2) is a 2-fold covering space of S O (3 ) . JR." is an oo-fold cover of the IHorus T il
JT (X I ,
•
.
•
, x ll )
=
=
SI
X . . . X
SI
c
e"
(exp[2JT ix I ] , . . . , exp[2JT i xn D
We shall now indicate how one can construct, in several ways, interesting covering spaces M for any manifold M that is not simply connected. (It will tum out that a simply connected M will have M itself as its only covering.)
21.2c. The Universal Covering Let Mil be a connected manifold. The universal covering manifold M n of Mn is constructed as follows : Pick a base point Po in M. A point of this new space M is then defined to be an equivalence class of pairs (p , y ) , where p is a point in M and y : [0, 1 ] -+ M is a path in M starting at" Po and ending at p, and where (p, y ) is equivalent to ( P I , Y I ) iff P = P I and the paths Y and Y I are homotopic. This last requirement means simply that the closed path Y Y I- I consisting of Y followed by the reversal of YI is deformable to the point Po . We then automatically have a covering map y ( l ) . To give a JT : M -+ M defined by assigning to the pair P , Y the endpoint P manifold structure to M we need to describe the local coordinate systems; we shall do this after the following simple example. We illustrate all this with M a 2-torus. =
THE FUNDAMENTAL GROUP AND COVERING SPACES
571
Figure 21 .6
The curves e and e' are homotopic, but neither is homotopic to e". Thus in our n ew space M, the universal cover of T 2 , the pair p, e and p, C ' will define the same point (to be described shortly) but p, e" will be represented by a different point p' . In the general case, we need to describe the manifold structure of M . We define a coordinate neighborhood of the pair p, Y on M by first taking a simply connected coordinate neighborhood U of p on M. Then to a point q in U we assign a curve consisting of the given Y followed by an arc Yp q in U from p to q . The homotopy class of Y Ypq is independent of the arc Ypq chosen since all arcs from p to q in U are homotopic as a result of the simple connectivity of U . Then a "lifted" neighborhood U of p, Y in M, by definition, consists of the classes of all such curves Y Y,'q for all q in U. This is illustrated in the toral case that follows.
p
Figure 21 .7
Since a pair q, Y Ypq is completely determined, up to homotopy, by the endpoint q , the points o f U described are i n I : I correspondence with the points q in U . Since U is a coordinate patch on M , we have succeeded in introducing local coordinates in the set U ; the local coordinates of q , Y Ypq in M are simply the local coordinates in U of q ! We do this for all p in M. By this construction, the map n: : M -+ M is such that each n: : U -+ U is a diffeomorphism. Because n: : M -+ M is locally a diffeomorphism, any Riemannian metric in M can be lifted by n: to yield a Riemannian metric in M, since the local coordinates in M yield the "same" local coordinates in M. By this construction, n: is also a local isometry, and of course the curvatures coincide at p and n: (P) . Let us verify that the universal cover of the torus T 2 is the plane ]R2 . To simplify our pictures, we shall consider new curves on the torus and illustrate with these.
572
BETTI NUMBERS AND COVERING SPACES
2B Plane
R?
B
PO�
PI O
P 20
Figure 21 .8 In the upper diagram we have drawn the torus in the usual way as a unit rectangle with opposite sides A identified and opposite sides B identified. We have drawn the closed curves A and B starting at the base point Po . In the lower ]R.2 diagram the point Poo corresponds to the pair Pa Y where y is the constant path whose locus is simply the point Po . We know that a simply connected patch around Po in T will be in 1 : 1 correspondence with a patch around Poo . As we move along the curve A from Po we also trace out a curve A starting out at oo P . On the completion of A in T we return to the point Po again. Since, however, the curve A in T is not homotopic to the constant curve Po, the pair (Po , y = Po) is not equivalent to the pair PoA . "ihis means that the endpoint P IO of A is not to be identified with its beginning point Poo ! For the same reason, the vertical line through P Ia is not to be identified with that through Poo . Likewise, if·one goes around A twice on T, in T we end not at Poo nor at P I O but rather at a new point P20 . The same procedure shows that on going around B we trace out a curve B that ends at a new point POI ' and so forth . We have also illustrated the case of a closed curve C in T that wraps twice around in the A sense and once in the B sense; its lift C in T ends at the point P2 1 . We also know, by definition, that any curve in T that starts at Poo and ends at P2 1 represents (i.e., projects down via n: to) a closed curve in T that is homotopic to C ! (x + n , Thus although T can be considered the plane with identifications (x , y ) y + m) , the universal cover T is the plane without identifications, that is, T = ]R.2 . �
573
THE FUNDAMENTAL GROUP A N D COVERING SPACES
Note that when M is simply connected, then by construction its universal cover M coin cides with M itself, since any pair of curves from Po to a point p are homotopic.
21.2d. The Orientable Covering This is the covering one obtains by using the same method as in the universal cover except that we now say that a pair p, Y is equivalent to a pair p, Yl iff when we transport an orientation from Po to P along Yl we obtain the same orientation as along y , that -1 i s, if when we translate an orientation along the closed curve Y YI we return with the ori ginal orientation. As in the construction of the universal cover, it is important that we are dealing with homotopy classes ; if a closed curve C preserves orientation, and if C ' is homotopic to C, then C' will also preserve orientation. If M is orientable, then the covering obtained reduces to M itself, but if M is not orientable we obtain a new space M . I n any case M i s called the orientable cover o f M, for, a s w e shall see, this M is always orientable. Consider, for instance, the Klein bottle, considered as a rectangle with the twisted identifications on the vertical sides
...- ... Po Po ....-_..:..:...--t *q A
K
B
B
Po L..-_----t..JPo A *(j''' POI
*7]"#
"q
*q' PIO
Poo
ibo B
Poo
*q
c
*Zj"
ho
Poo B
T
c
IR
'q'
Figure 21 .9
Poo
574
BETTI NUMBERS AND COVERING SPACES
I n the second diagram w e have indicated how one can view the Klein bottle a s the plane with twisted identifications ; the point q in K corresponds to all of the points q , qt , q" , . . . , in the plane. As we move along the curve A in K starting at Po, it is equivalent to mov ing along the segment POOP O in ll�? When we reach the point Po again in K we note th at We I have traversed a closed path A in K along which the orientation has been reverSed. This means that in our � 2 picture of K , the point P IO is not to be identified wi th p in our model for this new covering space. If, however, we traverse the curve A tWic the orientation is preserved; thus in the � 2 picture the point P2 is to be identi fied wi 0 Poo , but not to P I O . On the other hand, P30' corresponding to traversing A three times' is to be identified with P IO ' and so on. On traversing B the orientation is preserved; hence PO I is still to be identified with Poo. It will then follow that in this new covering K , horizontal lines are to be identified if they are separated by multiples of 1 unit, whereas vertical lines are to be identified (without twisting) if they are separated by multiples of 2 units. If we make such identific ations in � 2 we see that the resulting space is simply a torus T of twice the area of K. The two-sheeted-orientable cover of the Klein bottle is the torus ! We have drawn the torus in the last figure as a rectangle with the usual identifications on the boundary, and no other identifications, q =f. qt. C is the closed curve that covers A twice. By the same arguments, it can be shown in general that the orientable cover of M is either M itself, if M is orientable, or a 2-sheeted cover of M .
� th
21.2e. Lifting Paths
Tr
Let : M --+ M be any covering of the manifold M. M and M are locally diffeomorphic under the map Tr . The fiber over p , (p) , is a disconnected set of points. (It is useful to keep in mind the examples of the universal covering � 2 over T 2 , with fiber an infinite set of points, and the orientable cover T 2 over the Klein bottle K 2 with fiber a pair of points.) Let P be any point in this fiber. Let C be a curve in M starting at some p and ending at some q . Since M and M are locally diffeomorphic, there is a unique curve C that starts at P and covers C , (C ) = C . Its endpoint q by construction is at some point in the fiber (q ) . This defines the lift of. C to M that starts at If C is closed, q = p, it may be that C is not closed; that is, it may be that q =f. p. This occurs in the universal covering iff the closed curve C is not homotopic to the constant curve p; in the orientable cover it occurs when C is a curve that reverses orientation. These follow essentially from the definitions of these covers. (In our definitions we based everything at a base point Po , but it is not hard to see that we get similar behavior if we choose a new base point p.) Consider now the case o f the universal cover. Let y be any closed curve in M that starts and ends at It projects down to a closed curve y = CY) starting and ending at p = Since the closed curve y is a lift of y , it must be that the curve y is homotopic to the constant map p in M. As we deform y to the point p we may cover this deformation, using the local diffeomorphism by a deformation of y to the point We have thus shown that M is simply connected.
Tr- 1
Tr- I
Tr(p). p.
p.
Tr
p.
Tr
Tr,
THE FUNDAMENTAL GROUP AND COVERING SPACES
575
Furthermore, by definition of the universal cover, the points of the fiber Jr - 1 (Po) are in 1 : 1 correspondence with the distinct homotopy classes of closed curves in M starting at Po · Summarizing, we have shown
Theorem (21.16): The universal cover M of M is simply connected and the number of sheets in the covering is equal to the number of elements (the order) Oj Jr l (M) . If a manifold is not orientable, there is some closed curve that reverses orientation.
By the same type of reasoning as in (2 1 . 1 6) we have the following explanation of the
terminology that we have been using:
Theorem (21.17): The orientable cover of M is always orientable. The number of sheets is 1 if M is orientable and 2 if M is not orientable. 2 1.2f. Subgroups of 1rl (M) The orientable cover of M resulted from identifying two curves Y and Yl from Po to p iff the closed curve Y Yl- 1 preserves orientation, that is, if the homotopy class of Y Yl- 1 l ies i n the subgroup of Jr l (M) consisting of orientation preserving loops. Similarly, given any subgroup G of Jr J (M) , we may associate a covering space MG of M as follows: We again consider pairs p , Y , and we identify p , Y with p , YJ iff the homotopy class of the loop Y YI- J lies in the subgroup G . For example, when G is the identity 1 of Jr l (M), the covering is the universal cover, whereas if G is the subgroup of orientation-preserving loops the cover is the orientable cover.
21 .2g. The Universal Covering Group Let Jr : G ---+ G be the universal covering space of a Lie group G . We shall indicate why it is that G itselfis then a Lie group ! For example, 5U (2) , being a simply connected cover of 5 0 (3) , is the universal covering group of 5 0 (3 ) . A simpler example is furnished by exp : ffi. ---+ 5 1 sending e E lR to e i e . This is a homomorphism of the additive group of real numbers onto the multiplicative group of unit complex numbers. We have already seen in Section 2 1 .2b that this makes ffi. a covering manifold for 5 1 . Since lR is simply connected, it is the universal covering group of 5 J For identity in G we pick any point e E Jr - I (e) in the fiber over the identity e of G. If g is any point i n G w e define g- l as follows: g can be represented by a path g et) in G joining the base point e to the point g ( l ) = g := Jr (g) . Then the inverse path g- l (t) joins e t o g- J . This path can b e covered b y a unique path in G that starts a t e . It ends at some point in Jr - J (g - I ) and we define this point to be g - I . Let g and h be points in G ; they can be represented by paths (� and Ch joining e to g E Jr (g) and to h E Jr (h) . Consider the path Cg followed by the left translate gCh ; •
576
BETTI NUMBERS AND COVERING SPACES
since gC" starts a t g and ends a t gh, the composite path starts at e and ends at gh. Its unique lift that starts at e ends in ;r - l (gh) . This endpoint is defi ned to be gh. These basic constructions can be shown to yield the required universal covering group (see, e.g., [P, chapter viii]) . Note that since w e may lift the Lie algebra 3' = G (e) uniquely t o e , the unive rsal
cover of G has the same Lie algebra as G.
21.3. The Theorem of S. B. Myers : A Problem Set
A spiral curve in the plane can have curvature have Gauss curvature � 1 and infinite area?
� I
and infinite length. Can a surface in space
Let Mil be a Riemannian manifold and consider a geodesic C joining p to q. Then the first variation of arc length vanishes, L' (0) = 0, for all variations whose variation vector J = 8xj8a is orthogonal to T. Consider the second variation in this case, as given by Synge's formula ( 1 2.6)
We shall construct (n - 1 ) such variations as follows: Let e2 , e 3 , . . . , ell be orthononnal vector fields that are parallel displaced along C and orthogonal to C ; this is possible since T is parallel displaced also. Define the (n - 1 ) variation vectors
where f is a smooth function that vanishes at the endpoints p and q . We e l : = T and use the e's as a basis along C.
21.3(1) Show that for i
= 2, . . . , n we have for the i th variation vector
and
Suppose now that the Ricci curvature is positive Ric (T, T) ::: c
>
°
and choose for variation function f (s) = sin(;r s j L ) .
may put
THE THEOREM OF S. B. MYERS: A PROBLEM SET
21.3(2)
Show that
L" 1/ i =2
L 1. (0)
<
-
[
L 7T 2 (n - 1 ) -C L2 2
-
577
]
and conclude then that if the geodesic C has length L such that
L > 7T
[ en C ] 1 /2 - 1)
then C can not be a length-minimizing geodesic from p to q . 21.3(3)
What does this say for the round n-sphere of radius a i n ]Rn+l ?
Now suppose that M is geodesically complete; the theorem of Hopf-Rinow (2 1 . 1 0) tes sta that between any pair of points there is a minimizing geodesic. Let us say that a geodesic ally complete manifold has diameter f.. if any pair of points can be joined by a g eodesic of length :::: f.. and for some pair p , q the minimizing geodesic has length exactly f.. . We have proved
Theorem of S. B. Myers (21.18): A geodesically complete manifold M n whose Ricci curvature satisfies Ric(T, T) :::: c > 0 for all unit T has diameter :::: 7T [(n - I ) jC] I / 2 .
Corollary (21.19): A geodesically complete Mil with Ric(T , T) :::: c > 0 is a closed (compact) manifold. In particular its volume is finite. T) = K is simply the Gauss curvature. The 2-dimensional version was proved by B onnet in 1 85 5 . )
(In the case of 2 dimensions, Ric(T,
P R O O F : For a given p i n M the exponential map exp : Mp -+ M is a smooth p map of all of ]R" into M, since M is complete. By Myers's theorem the closed ball of radius r > 7T [( n - I ) j c] I / 2 in M (p) is mapped onto all of M. This closed ball is a compact subset of ]Rn and its image is again compact. D
21.3(4) The paraboloid of revolution z = x 2 + y 2 clearly has positive curvature (and can be computed from Problem 8 .2(4» and yet is not a closed surface. Reconcile this with (2 1 . 1 9) .
Now let Mil be geodesically complete with R i c (T T) :::: c > O. It is thus compact. Let M be its universal cover. We use the local diffeomorphism 7T : M -+ M to lift the metric to M, and then, since 7T is a local isometry, M has the same Ricci curvature. Every geodesic of M is clearly the lift of a geodesic from M, and so M is also geodesically c omplete. We conclude that M is also compact. We claim that this means that M is a finite -sheeted cover of M ! Take a cover { U, V, . . . } of M such that U is the only set ,
578
BETTI NUMBERS A N D COVERING SPACES
holding P o and U is s o small that i t is diffeomorphic to each-connected c o m pone nt of . . I ( U ) . The Inverse images of U, V , . . . form a cover of M , where eac h c on n Jr ee component of Jr - I ( U ) is considered as a separate open set. It is clear that i f M wted ere infinite-sheeted then any subcovering of M would have to include the infin ite CO llecti on in Jr - I ( U ) . This contradicts the fact that M is compact. From (2 1 . 1 6) we have -
Myers's Corollary (21.20): If Mil is complete with positive Ricci curvature bounded away from O. then the universal cover of M is compact and Jr l ( M ) is a group offinite orda Thus given a closed curve C in M, it may be that C cannot be contracted to a point, but some finite multiple k C of it can be so contracted. We have observed this before in the case M = lRp3 . This should first be compared with Synge's theorem ( 1 2. 1 2) . It is stronger than Synge's theorem in that (i) M needn' t be compact, nor even-dimensional, nor orientable. and (ii) positive Ricci curvature Ric ( e l e l ) , being a condition on a sum of section curvatures L j > 1 K (e l 1\ ej ), is a weaker condition than positive sectional curvature. On the other hand, Synge's conclusion is stronger, in that Jr l , being finite, is a weaker conclusion than Jr l consisting of one element. Synge's theorem does not apply to lRp3 whereas Myers 's theorem does (and in fact the fundamental group here is the group with 2 elements Z 2 ) , but Myers 's theorem tells us that even-dimensional spheres have a finite fundamental group whereas Synge tells us they are in fact simply connected. There is a more interesting comparison with Bochner's theorem ( 1 4.33). Myers's the orem is in every way stronger. First, it doesn't require compactness; it derives it. Second, it concludes that some multiple kC of a closed curve is contractible. Now in the process of contracting kC, k C will sweep out a 2-dimensional deformation chain C2 for which BC2 = kC see 1 3 .3a(III), and so C = B (k- I C2 ) . This says that C bounds as a real I -cycle, and thus b l (M) = O. Thus Myers 's theorem implies B ochner's. We have also seen in Section 2 1 .2a that contractibility is a stronger condition than bounding, for a l oop. Although it is true that Myers's theorem is stronger than Bochner's, it has turned out that Bochner's method, using harmonic forms, has been generalized by Kodaira, yielding his so-called vanishing theorems, which play a very important role in complex manifold theory. Finally, it should be mentioned that there are generalizations of Myers's theorem. Galloway [Ga] has relaxed the condition Ric (T, T) :::: c > 0 to the requirement that Ric (T, T) :::: c + df/ds along the geodesic, where f is a bounded function of arc length. Ric (T, T) need not be positive in this case in order to demonstrate compactness. Galloway uses this version of Myers 's theorem to give conditions on a space-time that will ensure that the spatial section of a space-time is a closed manifold! '
�
21.3(5 ) Distance from a point to a closed hypersuiface. Let V"- I be hypersurface of the geodesically complete Riemannian M il and let p be a point that does not lie on V. We may look at all the minimizing geodesics from p to q, as q ranges over V . The distance L from p to V is defined to be the greatest lower bound of the length s of th ese
THE THEOREM OF S. B. MYERS: A PROBLEM SET
579
geodesi cs. Let V be a compact hypersurface without boundary. Then it can be shown that this infimum is attained, that is, there is a point q E V such that the minimizing geodesic C from p to q has length L. Parameterize C by arc length s with p = C (O) .
Figure 21 .1 0
(i) Show from the first variation formula that C strikes V orthogonally. (This gener alizes the result of Problem 1 .3(3). ) (ii) Consider a variation vector field of the form J (s) = g (s )e2 (s) where e 2 is parallel displaced along C and g is a smooth function with g (O) = 0 and g eL) = I . Then LI/ (O) is of form B(J, J) + J;� { [ g'(S) [ 2 - [ g (s) [ 2 R 2 1 2 1 Ids , where B (J, J) is the normal curvature of V at the point q for direction J (L) and hypersurface normal T (L) ; see 0 1 .50) . By taking such variations based on (n - 1 ) parallel displaced orthonormal e2 , . . . , en , all with the same g, and putting g (s) = s/ L, show that
t L ;' (O) ;= 2
=
H (q) +
(n
-
L
1)
_
(�) Jor L s 2 L
Ric ( T , T)ds
where H (q) is the mean curvature of V at q for normal direction T. (iii) Assume that M has positive Ricci curvature, Ric (T , T) ::: 0 (but we do not assume that it is bounded away from 0) and assume that V is on the average curving towards p at the point q ; that is, h : = H (q) < O. Show then that our minimizing geodesic C must have length L at most (n - 1 ) / h . In general relativity one deals with timelike geodesics that locally maximize proper time (because of the metric signature - , + , + , +). Our preceding argument is similar to analysis used there to prove the Hawking singularity theorems, but the pseudo Riemannian geometry involved is really quite different from the Riemannian and forms a subj ect in its own right. For further discussion you may see, for example, [Wd, chaps. 8 and 9 ].
580
BETTI NUMBERS AND COVERING SPACES
21 .4. The Geometry of a Lie Group
What are the curvatures of a compact group with a bi-invariant metric?
21 .4a. The Connection of a Bi-invariant Metric Let G be a Lie group endowed with a bi-invariant metric. (As we know from Theor e (2 1 .8), such metrics exist on every compact group, and of course on any commutati group. The plane G = ]R.2 can be considered the Lie group of translations of the p lan e itself; (a , b) E ]R.2 sends (x , y) to (x + a , y + b). This is an example of a nonc ompac t Lie group with bi-invariant metric dx 2 + dy 2 .) To describe the Levi-Civita connection V x Y w e may expand the vector fields in terms of a left invariant basis. Thus we only need V x Y in the case when X and Y are left invariant. From now on, all vectorfields X, Y, Z , . . . will be assumed left invariant. We know from Theorem (2 1 .9) that the integral curves of a left invariant field are geodesics in the bi-invariant metric, hence Vx X = O. Likewise
:
(2 1 .2 1) that is, 2Vx Y = [X, Y]
exhibits the covariant derivative as a bracket (but of course only for left invariant fields). Look now at the curvature tensor
R (X, Y ) Z = VxVyZ- VyVxZ- V [X,YjZ In Problem 2 l .4( 1 ) you are asked to show that this reduces to
R (X, Y ) Z = For sectional curvature, using (20.35),
-4 ( R (X , Y)Y, X )
= =
-l [ [X, V] , Z]
(2 1 .22)
( [[X , V] , Y] , �) - ( V, [[X, V ] , X ] ) ( Y, [X, [X, Y] ] ) = - ( [X, V] , [X, V] ) =
or
K (X 1\ Y)
=
I I [X, Y] 1 1 2
(2 1 .23 )
Thus the sectional curvature is always ::: 0, and vanishes iff the bracket of X and Y
vanishes!
For Ricci curvature, in terms of a basis of left invariant fields e j , . . . , en l � . Rlc (e j , e j ) = � � K (e l l\ ej ) = �
4 I
I
[ e j , ej ] 2 j>l j Thus Ric( X, X) ::: 0 and = 0 iff [X, Y] = 0 for all Y E � . The center of the Lie algebra is by definition the set of all X E � such that [X, Y] = 0 for all Y E � . Thus if the center of W is trivial we have that the continuous -
THE GEOMETRY OF A L I E G ROUP
581
fu nction X t--+ Ric(X, X) is bounded away from 0 on the compact unit sphere in W at the i dentity. But since the metric on G is invariant under left translations, we then concl u de that the Ricci curvature is positive and bounded away from 0 on all of G . From Myers's theorem we conclude
Weyl's Theorem (21.24): Let G he a Lie group with hi-invariant metric. Sup
pose that the center of W is trivial. Then G is compact and has afinitefundamental gro up Jr l (G). This improves (2 1 . 1 3) since it can be shown that if there is no I -parameter subgroup in the center of G then the center of �/ is trivial; see Problem 2 1 .4(2). Note also that the condition "the center of �' is trivial" is a purely algebraic one, unlike the condition for the center of the group appearing in Theorem (2 1 . 1 3).
21.4h. The Flat Connections
We have used the Levi-Civita connection for a bi-invariant Riemannian metric. When
such metrics exist, this is by far the most important connection on the group. On any group we can consider the flat left invariant connection, defined as follows : Choose a basis e for the left invariant vector fields and define the connection forms w to be 0, Ve = O. (There is no problem in doing this since G is covered by this single frame field.) Thus we are forcing the left invariant fields to be covariant constant, and by construction the curvature vanishes, dw + w 1\ W = O. This connection will have torsion; see Problem 21.4(3). Similarly we can construct the flat right invariant connection.
P roblems
21 .4(1 ) Use the Jacobi identity to show (21 .22) . 21 .4(2) Suppose that X is a nontrivial vector in the center of �/ ; thus ad X( Y) = 0 for ali Y in w . Fill in the following steps, using ( 1 8.32), showing that etX is i n the center of G. First etad X y Y. Then e txYe-tX Then etX i s i n the center of G. =
=
Y. Thus exp(etxYe-tx )
= eY .
21 .4(3) Show that the torsion tensor of the flat left invariant connection is given by the structu re constants
TJk -Cjk· =
CHAPTER
22
Chern Forms and Homotopy Groups How can we construct closed p-forms from the matrix 8 2 of curvature forms?
22. 1 . Chern Forms and Winding Numbers
22.1a. The Yang-Mills "Winding Number" Rec all that in (20.62) and (20.63), we were comparing, on a distant 3-sphere S 3 the interior frame eu with the covariant constant frame ev ,
C
]R4,
eu (x ) = ev (x ) g v u (x ) g v u : S 3 -+ S U (n)
the gauge group being assumed S U (n) . We saw i n (2 1 . 1 4) that the Cartan 3-form on S U (n)
Q 3 = tr g - l dg 1\ g - l dg 1\ g - l dg is a nontrivial harmonic form, and we now consider the real number obtained by pulling this form back via g v u and integrating over S 3 (22. 1 ) We shall normalize the form Q 3 ; this will allow u s to consider (22. 1 ) as defining the degree of a map derived from g v u . Consider, for this purpose, the SU (2) subgroup of S U (n)
SU (2) = S U (2) x In - 2 : =
[ SU (2) 0
�J
In
C
S U (n)
The Cartan 3-form Q 3 o f SU (n) restricts to Q 3 for SU (2) , and w e shall use as normal ization constant
which we proceed to compute.
583
584
C H E R N F O R M S A N D H O M OTOPY G R O U PS
Q 3 and the volume form vol 3 on S U (2) , in the bi-invariant metric, are both bi invariant 3-forms on the 3-dimensional manifold S U (2) ; it is then clear th at r. . · ' 3 1S some constant multiple of vol 3 • From ( 1 9.9) we know that ia, I,Ji, i a21../'5., and ia3 / ,Ji form an orthonormal basis for �LL(2) with the scalar product (X, Y) = tr X Y (recall that ( 1 9.9) defines the scalar product in i � , not 3') ' Then, from (2 1 . 5 ) and ( 1 9.6) -
S ince the i a 's/ ,Ji are orthonormal, we have vo I 3 ( i a l ia2 , i (3 ) = 23 / 2 . Thu s shown ,
we h ave
(2 2. 2) What, now, is the volume of S U (2) in its hi-invariant metric? S U (2) is the unit sphere S3 in ([ 2 = �4 where we assign to the 2 x 2 matrix u its first column. The identity element e of S U (2) is the complex 2-vector ( 1 , 0) , or the real 4-tuple N = ( I , 0, 0, 0) T . The standard metric on S 3 C �4 is invariant under the 6-dimensional rotation group S O (4) , and the stability group of the identity is the subgroup 1 x S O (3) . Thus S 3 = S O (4) / S O (3 ) . The standard metric is constructed first from a metric in the tangent space S� to S 3 at N that is invariant under the stability group S O (3) and then this metric is transported to all of S3 by the action of SO (4) on S O (4) I S O (3) . Since the stability group S O (3) is transitive on the directions in S� at N, it should be clear that this metric is completely determined once we know the length of a single nonzero vector X in S� . Of course S U (2) acts transitively on itself SU (2) = S 3 by left translation. It also acts on its Lie algebra S; by the adjoint action ( 1 8 . 3 1 ), and we know that the bi invariant metric on S U (2) arises from taking the m�tric (X, Y) = tr X Y at e and left translating to the whole group. Now the adj oint action of S U (2) on S; is a dou ble cover of the rotation group S O (3) (see Section 1 9 . 1 d) and thus is transitive again on directions at e. We conclude then that the bi-invariant metric on S U (2) = S 3 is again determined by the length assigned to a single nonzero vector in S; = S� . The hi-invariant metric on S U (2) is simply a constant multiple of the standard metric on S3 . Consider the curve on S U (2) given by diag(e i li , e - i ll ) ; its tangent vector at e is simply i a3 whose length in the bi-invariant metric is ,Ji. The corresponding curve in ([2 is (e i ll , 0) T , which in �4 is (cos e, sin e, 0, 0) T , whose tangent vector at N is (0 1 0 0) T with length 1 . Thus the bi-invariant metric is ,Ji times the standard metric on the unit sphere S 3 . Since a great circle will then have bi-invariant length 2:rr ,Ji , we see that the bi-invariant metric is the same as the standard metric on the sphere of radius "fi. (Note that this agrees with the sectional curvature result (2 1 .23), K Cial 1\ i (2 ) = ( /4) I I [ia l , i a2 ] 11 2 I I I ia l 1\ ia2 11 2 = ( /4) II 2 ia3 11 2 I I ia l 1\ ia2 11 2 = 1 /2.) The volume of the unit 3-sphere is easily determined. -
-
CHERN FORMS AND WINDING NUMBERS
585
Figure 22. 1
Thus our sphere o f radius .fi has volume (2 3 / 2 ) 2n 2 , and so
j
SU (2)
Q 3 = 24n 2
Finally we define the winding number at infinity of the instanton by
1 j g vu Q 3 = -1 1 -2 24n 2 = �1 24n *
24n
S3
g" u (S')
g" u (S3 )
Q3
(22.3)
tr g - I dg 1\ g- l d g 1\ g - I dg
This is the degree of the map g vu in the case when G = S U (2) . What it means in the case S U (n) will be discussed later on in this chapter.
22.1b. Winding Number in Terms of Field Strength Chern's expression (20.68) in the U ( 1 ) case suggests the possibility of an expression for this winding number in terms of an integral of a 4-form involving curvature. We shall assume that the Y-M potential Wu is globally defined in U ; that is, Wu has no singularities in U, j ( e u ) = O. Consider the following observation, holding for the curvature 2-form matrix for any vector bundle over any manifold: e
1\ e
= (dw + w
1\ w) 1\
(dw
+ w 1\ w)
= dw l\ dw + dw l\ w l\ w + w l\ w l\ dw + w l\ w l\ w l\ w
Use now tr( w 1\ w 1\ dw) = tr(dw 1\ w 1\ w) and, as in Theorem (2 1 .3) tr(W I\ W l\ w l\ w) = 0
586
CHERN FORMS AND HOMOTOPY GROUPS
Then
tr e /\ e
=
d
/\
tr (w /\ dw) + 2 tr(dw w /\ w)
Also
w
de /\ W /\ w)
=
dw /\
and so
w w - W /\ dw /\ w + w /\ w /\ dw /\
=
/\
/\
dtr(w w /\ w) 3tr (dw /\ w w) Thus we have shown
Theorem (22.4): For any vector bundle over any M il we have tr e e /\ e) = d tr
{w
/\
dw +
/\
�w w /\ w}
Thus tr e /\ e is always locally the differential o f a 3-form, the Chern-Simons 3-form. Of course is usually not globally defined. Now back to our Y-M case considered in Section 20.6a. In that case e vanishes on and outside the 3-sphere S 3 , and so
w
w /\ dw = W /\ (e
on and outside S 3 . Then from (22.4)
But
wu
- w w) /\
=
-w /\ W /\ W
! tr e /\ e = la u =s3 - �3 tr /\ w /\ w U
=
w
g- l dg on S 3 ; see (20.6 1 ) . (22.3) then gives
Theorem (22.5): The winding number of the instanton is given by
r tr wu wu /\ wu -� r tr e � 24n } 8n J�4 S3
/\
=
/\
Note that tr e /\ e is not the Lagrangian, which is basically tr e
F /\ F
= =
(F /\ F) 0 1 23 dt /\ dx /\ dy /\ dz
e /\
*e
L L E ij kl Fij Fk1 dt /\ dx /\ dy /\ dz
(22 .6)
i < j k
whereas
F /\ *F = L Fj k F jk dt /\ dx /\ dy /\ dz j
We have j ust shown that the winding number of an instanton is given, in tenn s of the Hilbert space scalar product (20.40), by (8n2) - 1 (e , *e) ; this scalar product is defined since e is assumed to have compact support. This is the degree of the map g : S3 SU (2) defined by the instanton. This degree is interesting for the following
-+
587
CHERN FORMS AND WINDING NUMBERS
reason: The Y-M fields are critical points for the Y-M action functional. In particular, a connection w yielding a (relative) minimum for S will be a Y-M field. But by Schwarz's i nequality (in the euclidean metric), the euclidean action S on M 4 will satisfy
8n 2 I
deg(g)
1 = 1 ( e , *e ) 1 ::::: 11 e 11 11 *e 11 = 11 e 11 2 = 2S
since * is an isometry on forms . Thus the degree yields a lower boundfor the euclidean action! Furthermore, we have equality iff *e is proportional to e . Now * * ex = ex ,
when ex is a 2-form. It is easily seen that * acting o n our 2-forms in M 4 is self-adj oint in the scalar product (20.40). Thus * has eigenvalues ± I on the 2-forms and so *e is proportional to e only when *e = ±e , that is, iff the connection is self-dual or anti-self dual; see (20.58). In particular, the self-dual fields with degree n and the anti-self-dual fields with degree - n will both yield Y-M fields having minimum action among all fields of degree ±n.
22. 1c. The Chern Forms for a U(n) Bundle
The topological significance of tr e /\ e, generalizing Poincare's theorem for closed surfaces, II KdA = 2n x (M 2 ), was discovered by Chern and will be discussed later in this chapter. tr e /\ e is but one of a whole family of significant integrands, the Chern forms. We shall define these forms now and then proceed to the topological questions in our remaining sections. Let A be any N x N matrix of complex numbers operating on complex N -space V = (f . Consider the characteristic (eigenvalue) polynomial for A det(A I - A) = (A - A d (A - A2 ) ' " (A - AN )
N l N = A - (A I + . . . + A N ) A - + . . . ± (A l A 2 . . . A N )
Putting A = - I yields
L [ tr 1\ A ] N
det(/ + A ) =
P
1' =0
(22.7)
(
= I + (tr A ) + tr where
2
1\
)
A +...+
N
( tr 1\ A ) (22.8)
tr tr tr
2
1\ A : = L A; Aj 3
i <j
N
i < j
1\ A : = L
Ai A j A,
1\ A := A l A 2 . . . AN = det A
are the elementary symmetric functions of the eigenvalues of A. The reason for this n otation is as follows: if A : V ---+ V then we may let A act on each of the exterior
588
CHERN FORMS A N D HOMOTOPY GROUPS
power spaces I\ P V b y the (linear) exterior power operation I\ P A
( 1\P A ) (v
1
/\ V2 /\
•
•
•
/\ V
p)
:
Av1
=
/\
A V2 /\
•
•
•
/\
A vP
We then take the usual trace of I\ P A on the space I\ P V . For example, 1\ N V is I -dimensional vector space and from (2.50)
N
( 1\ A ) (v 1
/\ V2 /\
. . . /\ v N )
= det A ( v 1
/\ V2 /\
•
•
•
/\ V
a
N)
and so tr 1\ N A = det A = A , . . . A N . Note that (A� + . . . + At) = tr A k is simply the trace of the kth (ordi nary matrix ) power of the matrix. Thus, for example,
(22.9) =
1 2
[ (tr A) 2
,
-
tr A 2 ]
k
In a similar manner it can be shown, using "Newton's identities," that each tr 1\ A can be expressed as a polynomial in tr A , tr(A 2 ) , tr(A k ) . We shall return to this point in a moment. Now let E be a complex e N bundle with structure group U (N) , base manifold Mn , and connection w . Consider the result of formally substituting for A in (22.7), the matrix of curvature 2- forms e = e u multiplied by i /277: •
det
•
•
(I + �:)
" Thus we are looking at a matrix whose " entry is 1 + (i /277: )(3" " and whose nondiagonal " fJ entry is (i /277: )(3" fJ and where we expand out the determinant in the usual way with products being replaced by /\ products; since e j k is a 2-form there is no problem w ith ordering. The result is a sum of forms of different degrees det 1 +
(
-277:i e )
where Cr (E) is a 2r-form on U The form c , is familiar
= 1 + := 1 C
-277:i
tr e
+...
+ C l (E) + c 2 ( E ) +
..
(22. 1 0) . + cN (E)
Mil , the rth Chern form.
i c, = __ tr e 277:
(22. 1 1 ) _i_ e " " 277: and in the case of a complex l ine bundle, ee< " is simply the 2-form e appearing in Theorem ( 1 7.28). For the tangent complex line bundle to an oriented surface, c, ( T M 2 ) = ( l /277: ) Kd A . For C2 , from (22.9) w e have (22. 12 ) =
C H E R N FORMS A N D W I N D I N G N U M B E RS
589
Suppose that the bundle actually has the special unitary group SU (N) for structure g roup, rather than U (N) . Since the Lie algebra then consists of traceless skew-hermitian matrices, tr e = 0, and thus in this case
c l (E) and furthermore
=
0
I
tr ee /\ e ) S JT z This is precisely the 4-form appearing in the winding number of an SU(2) instanton,
Cz (E)
=
g iven in (22.5) ! In the general case, note that the matrices e are only locally defined, and in an overlap (}v = c v u eu cv � ' However det
( I + i2JT ) ev
=
det
{I + i
2JT
= det c v u
_I
c v u eu c v u
(I + �: )Cv�
} = det
Cr
(I + �:)
shows that each Chern form cr (E) is in fact a globally defined 2r-form on all of Mn ! In Problem 22. 1 ( 1 ) you are asked to show that each is a real form. We can see that is a closed 2-form as follows: From -2JT idc l = d tr e = tr d e , and from Bianchi this is tr ee /\ w - w /\ e ) . But tr w /\ e = tr e /\ w since e is a = 0, as claimed. It is even simpler to remark that locally 2-form. We conclude that () = d w + w /\ w and then tr e = d tr w
CI
dCI
since tr w /\ w = - tr w /\ w = O. Thus tr e i s locally exact, hence closed. For an SU (N) bundle, Cz is locally the differential of the Chern-Simons 3-form given in Theorem (22.4), and so is a closed 4-form in this case. We can also see this directly for any U (N) bundle, from the Bianchi identity. From (22. 1 2)
Cz
- ( S JT 2 )dcz
=
d [ tr e
/\
tr e - tree
/\
e)] =
-d tr ee /\ e )
But, from ( I S.46) and (20.55), d tr ee /\ e ) = tr V ( e /\ e ) = 0, since v e = O. A s w e have mentioned (but not proved), Newton 's identities show that each Cr is a polynomial in forms of the type tr e e /\ e /\ . . . /\ e ) ; we have shown this for C I and Cz and you are asked in Problem 22. 1 (2) to verify it for (For a derivation of the Newton identities, see [Ro, ex. 1 , p. 1 32] , but not before reading the remainder of this section.) Since V of such a polynomial vanishes by Bianchi, we conclude that each Chern form is closed. We present a different proof of this important fact now.
C3.
Cr
Theorem of Chern and Weil (22.13): Each is a closed 2r-form and thus de fines a real de Rham class. Furthermore, different connections for the U (N) bundle will yield Chern forms that differ by an exact form and hence define the same de Rham cohomology class. PROOF :
We sketch briefly a proof from Roe's book [Ro, p. 1 1 3 ] .
590
CHERN FORMS AND HOMOTOPY GROUPS
power series expansion s . For exam ple , th e m trj a ce = has a form + q e considered previously, where q = i nverse. If we write e r : = e 1\ e 1\ 1\ e , factors We shall look a t
I
a = a )
formal
�
i / 2rc ,
.
•
r
.
(22. 1 4) Thi s makes sense si nce it is only a finite series, e
1\ e 1\ . . . 1\ e
vanish ing Whe n the
number of factors exceeds half the dimension of the manifold that we let the connection
w
M.
vary smoothly with a real parameter
Suppo se now
t, w
=
w et) .
t . B u t for any nonsi ngular matrix a (t) we have for the derivative of its determinant 1 a (t) 1 da j k d 1 a C t) 1 =L dt aaj k dt j k = A aj k = 1 a 1 (a - I ) kj a j k = 1 a 1 tr[a - 1 a] j where A k i s the signed cofactor o f a jk . Hence Then both the curvature e and the matrix a vary with
[ �]
d log
, 1 a Ct) 1 ---- = tr [a - I a] dt Thus, putting e = dw + w 1\ w, e = dill + w 1\ ill + ill 1\ w , a = qe d log 1 a C t) 1 I '" = L ( - ) q r + 1 tr [ e ' 1\ (dill + w l\ ill + ill l\ w) ] dt
(22. 1 5)
r
One sees immediately by induction from B i anchi that
for
r
2: 0, with eO
=
Furthermore, tr[e r
degree. Hence
l.
de '
= er 1\
1\ ill 1\ w ] =
w - W 1\ e r
- tr [ w
1\ e r 1\ ill ] ,
d log 1 a Ct) 1 " ( - I ) ' r + 1 q =� dt r
or
trW
since er
1\ ill
is a form of odd
1\ dill + d e r 1\ ill ]
d l og 1 a Ct) 1 = d L ( - 1 )'" q r + 1 tr[e' 1\ ill ] dt
( 22 . 1 6)
r
d log 1 a Ct) 1 /dt as the differential of a s u m o f forms ( of various degrees). Note also that the forms on the right are i ndeed globally definedforms on the base space M, since both e r and ill are forms of type Ad G ; thi s was Problem 1 8.3(4). exhibits
As a first consequence of
(22. 1 6) note the following: If w and w' are two connections on M, then Problem 20.3 ( 1 ) shows that their convex combination w et) = tw + ( l - t ) w' is again a connection. This gives a line in the affine space of all connections on
M that starts at w' and ends at w. Now the flat connection w = 0, e = 0, is not necessaril y a connection on M for the given bundle (why ?), but it is a connection on a single coordinate patch U of M. Then w et) = tw i s a line o f connections o n U j oining any given connection w = w ( l ) t o the flat connection w (o) = 0. Since a (O) = I, we have, from (22. 1 6) , log
1
1 + qe
1= d
{
11 �
( - l r q r + 1 tr [ e r (t ) 1\ w]
}
dt
591
HOMOTOPIES AND EXTENSIONS
and so log 1 I + qa 1 is locally exact (being exact on U) hence closed; in fact it is of the form log 1 I + q8 1= df3 where
f3
:=
But then
l' { [q tr w - q 2 tr a Ct)
/\
w + . . · ] }dt
1 1 I + qa 1 = exp log 1 I + q8 1 = exp df3 = 1 + df3 + -df3 2!
/\
df3 + . . .
is again locally exact, except for the constant term, hence closed. We are finished with the first part of Theorem (22. 1 3) . Consider now a pair o f global connections w and W i o n M and the line tw' + ( 1 - t)w in the space of connections. From (22. 1 6) we have log 1 aw, 1 - log 1 aw 1 = dy for a globally defined form y on M
y = r ' 2:) - l yqr+, tr[ar ( t)
Jo
r
/\
( Wi -
w) ]dt
Then
1 a w, 1 = exp{log 1 aw, 1 - log 1 aw I } = exp dy 1 aw 1
--
1 2!
= 1 + dy + - dy
/\
dy +
...
=:
1 + dv
and so 1 a w, 1 - 1 aw 1 = 1 a w 1 /\dv. But we have just seen that 1 aw 1 = det(I + qa) i s closed. Hence 1 aw, 1 - 1 a w 1 is globally exact, proving the second part o f the theorem. 0 P roblems -------
22.1 (1 ) Show di rectly from det( I + ie /2:rr ) that each Cr is a real form when the structure group is a subgroup of U( N) .
22.1 (2) Express C3 as a polynomial i n tr e , tree /\ e ) , and tree /\ e /\ e ) .
22.2. Homotopies and Extensions
Is SU (n) simply connected?
22.2a. Homotopy In Section 1 0.2d we discussed when two closed curves in M are homotopic. We now introduce the general concept of homotopic maps. Let fo and f, be two maps of a space W into Mn . We say that they are homotopic if there is a map F : W x I --* M of the "cylinder" W x [0, 1 ] into M such that
F(w, O)
=
fo (w) and F (w , 1 ) = f, (w)
592
CHERN FORMS AND HOMOTOPY GROUPS
F o
=:> w
c
___
Figure 22.2 Each of the maps ft , defined by f, (w ) : = F ( w , t ) , is homotopic to the "origi nal" map fo . If fl maps all of W into a single point Po we say that fo is homotopic to the
constant map Po .
We shall be especially concerned with the case when W = Sk is the unit k sphere k = 0, 1 , 2, . . . , in ]Rk + I , even when k > n = dim M ! S k is of course the boundary of the closed (k + I )-ball D k + 1 and the following simple observation will play a cruc ial role in our final section. -
Extension Theorem (22.17): f : S " ---+ f can be extended to a map of the ball
Mn
,
is homotopic to a constant map iff
Suppose that I' : D k + 1 ---+ M extends f : Sk ---+ M; thus I' (x) = f (x) for I I x I I = 1 . Define F : Sk x I -+ M by
PROOF :
F (x, r)
=
f' { ( 1
-
r)x},
Then F (x, 0) = I' (x) = f (x) and F (x, 1 ) = I' (0) shows that f i s homotopic to the constant map 1 ' (0) . Suppose, on the other hand, that f ( = fo ) is homotopic to the constant map fl (x) = Po E M . Then we have a map F : Sk ---+ M with F (x , 0) = f (x) and F (x 1 ) = fl (x) = Po . Define an extension I ' : D k + 1 ---+ M by I' (rx) = F (x, 1 r) for 0 :::: r :::: 1 . The extension theorem i s important when discussing defects, see [Mi] . 0 ,
-
22.2h. Covering Homotopy
Let JT : E ---+ Mil be a vector bundle and let f : W ---+ E be a map of a space W i nto the bundle space E . Then we get a map [ : W ---+ M n into the base space by [ : = JT 0 f ·
593
HOMOTOPIES AND EXTENSIONS
M
Po
Figure 22.3
Suppose now that we have a homotopy F of [ to a new map L : W � M . We claim that we can "cover" this homotopy by a homotopy of the original map f; that is, there is a map F : W x I � E such that F ( w , 0) = few) and T{ F ( w , t ) = F ( w , t). A sketch goes as follows: Let the vector bundle T{ : E � M have a connection. Consider a fixed point w E W and look at the curve C : t � F ( w , t) in M. There is a unique lift of this curve to a curve C in E starting at f (w) that represents parallel translation along C. In other words, we look at the unique curve in E that starts at f ew), lies over {2, and is tangent to the n -plane distribution defined locally by d1jra + wa fi 1jr fi = 0
Note that if [ is homotopic to the constant map
Po (as in the second part of our figure)
it need not be that f will be homotopic to a constant map; the points F ( w , 1 ) of the lifted homotopy will lie on the fiber T{ - I (Po) but will not necessarily reduce to a single
point in the fiber. What we have said for a vector bundle can also be shown to hold for a principal fiber bundle. The lifted curves are then tangent to the n-plane distribution w* = g - I wg + g - I dg = 0 It turns out that one can cover homotopies in any fiber bundle, without any use of a connection. In fact, one generalizes the notion of a fiber bundle to that of a fiber space; this is a space P and a map T{ : P � M such that homotopies can always be covered, as defined earlier. Such spaces need not be local products.
594
C H E R N FORMS A N D HOMOTOPY G ROUPS
22.2c. Some Topology of SU(n)
5 U (n) is represente � by n x n matrices acting on (C" . S ince each g E 5 U (n) i s unitary, 5U (n) sends the umt sphere 5 2n - 1 C (C" 5 2n - 1 = {z E en I 1 Z I 1 2 + . . + 1 Zn 1 2 = I } into itself. It is clear that 5 U (n) acts transitively on 5 2n - 1 , for the point ( 1 , 0, . . . , 0) can be sent into the point Z = , Zn ) simply by writing down some g E S U (n) T having Z as its first column. The isotropy subgroup for the point ( 1 , 0, . . . , 0) i s clearly .
the subgroup
(Z I
•
.
.
.
[�
� ]
5U( - 1) which we shall briefly denote simply by 5 U (n - 1 ) . 5 U (n) 5 2n - 1 = (22. 18 ) 5 U (n - 1) and in fact 5 U (n) is a principal 5U (n - 1 ) bundle over 5 2n - 1 (see Theorem ( 17. 1 1» . If P is a fiber bundle over M with fiber F we shall write symbolically (22. 19 ) and we shall frequently omit the projection map JT . Thus we write
5U (n - 1 )
-+
5 U (n)
-+
52n - 1
Theorem (22.21): If F -+ P -+ M is a fiber bundle with connected M connected F, then P is connected.
(22. 20) and
Let P and Po be points in P . Project them down to points JT (p) and ) in M ( JT Po . Since M is connected there is a curve in M joining these two points.
PROOF :
p
F
Po
n(p)
----�.�---..�----4·-- M
Figure 22.4
This curve can be considered a homotopy from the constant map of a poin t w into JT (p) , to the constant map of the point w to JT ( Po ) . Cover this homotopy by a
595
HOMOTOPIES AND EXTENSIONS
p ath from P to the fiber through Po, that is, from P to some point P I in this fiber. Since this fiber is assumed connected we can find a curve in this fiber from P I to po. We have joined P to P o by a succession of two paths in P . D
Corollary (22.22): S U (n ) is connected. PROOF:
S U ( 1 ) is a single point. SU (2) is a 3-sphere and is connected, as are all k-spheres for k > O. From SU (2) � S U (3) � S5 we see that S U (3) is connected. Induction gives the corollary. D
See Problem 22.2( 1 ) at this time. Recall that we say that M is simply connected provided every map of a circle into M is homotopic to a constant map. During the homotopy, the closed curve gets "contracted" or "deformed" to the point.
Theorem (22.23): Let F � P � M be a fiber bundle whose fiber F and base M are simply connected. Then P is simply connected. PROOF:
Let C be a closed curve in P . Project it down to a closed curve n ( C) in M. Since M is simply connected, n (C ) can be contracted to a point Po in M . We may cover this homotopy by a deformation of C into the fiber over Po ; that is, C is deformed into a new closed curve lying in the fiber n - I (Po) . Since the fiber is simply connected, this new closed curve can be shrunk to a point in the fiber. Thus the composition of the two deformations deforms C to a point, as desired. D
P roblems
22.2(1 ) Show that S O(n) is connected. 22.2(2) We know that the cartesian product of connected manifolds is connected; this is the special case of (22 .2 1 ) when F ---+ M x F ---+ M is simply a product bundle. I n a prod uct bundle we also have the converse (which is evident from a picture); if M and M x F are connected, then F is connected. That this need not be true when M x F is replaced by a twisted prod uct, that is, a bundle P, may be seen as follows: Denote the principal frame bundle to a Riemannian 3-man ifold M3 by 0(3) ---+ F M ---+ M. 0(3) is defin itely not connected , being the disjoint union of S O(3) and those 9 E 0(3) with det 9 - 1 . I n spite of this, show that if M is connected and not orientable, then F M is connected! I n particular FM in this case i s not a prod uct. A simpler example Z 2 ---+ S 1 ---+ S 1 is the 2-fold covering of a circle by itself. Show that this is realized in the case of the unit normal bundle P to the central circle S 1 of the (infin ite) Mobius band Mo. =
22.2(3) Show that SU(n) is simply connected.
596
CHERN FORMS AND HOMOTOPY GROUPS
22.3. The Higher Homotopy Groups
7rk(M)
Why is the alternating sum of Betti numbers equal to the Euler characteristic?
22.3a. 7rk(M)
We shall consider continuous maps f : S k ---+ M of a k-sphere into Mil . We shall always ask that some distinguished point on Sk , the "north pole," be sent into a dis tingu ished base point, written * in Mil . We shall only consider k ::: 1 . For technical reasons w. e consider Sk to be the unit k-cube, J k = [0, 1 ] x . . . x [0, 1], with the entire boundary J k identified with a single point, the north pole .
[I
1 -•
--
-:--
-
--111--
'I
_____
.....
...1-- ' I
Figure 22.5
Then f : S k ---+ M is a map f : J k ---+ M such that f (i) = * . In our diagrams the heavy portions are always mapped to *. To say that fo and fl are homotopic, fo '" f" is to say that there is a map F : J k x J ---+ M such that F (y , 0) = fo (y ) , F (y , 1 ) = II (y), and F (north pole, t) = * , 0 :s 1 :s I
Figure 22.6
(Again the heavy portions are sent into the base point.) We compose two maps f : Sk ---+ M and g : Sk ---+ M using the first coordinate, as we did for loops, but this time the result is written f + g : 1
< < 1, 02
1
- < 2 -
< 1 I, -
THE HIGHER HOMOTOPY GROUPS
7rk(M)
597
f
briefly
g
Figure 22.7
Again, two maps are to be identified if they are homotopic. The homotopy classes of such maps define
the kth homotopy group
(It can be shown that if f' J and g ' J + g . ) The identity is g then f' + g ' represented by maps homotopic to the constant map J = * , and the inverse of the map J (t l , " " fn ) is represented by J ( 1 - f l , f2 , " " fn ) ' The composition is written additively since these classes of maps form a commutative group (if k ::: 2) . The commutativity can be "seen" from the following sequence of homotopies where a �
f
g
squash
f *
g
�
�
*
g
f
*
*
g
*
f
f
g
*
g
f
Figure 22.8
whole box labeled * is to be sent into the base point. See [H,Y] for details. Note that this procedure will not work in the case n = 1 ; there is no room to maneuver. This is why the fundamental group lr 1 can be nonabelian.
22.3b. Homotopy Groups of Spheres lrk (sn ) consists of homotopy classes of maps of a k-sphere into an n-sphere. We have al ready discussed lrl ( S l ) = Z where the homotopy class is characterized by the Brouwer
degree of the map. (We have shown that maps of different degrees are not homotopic, but we have not proved the converse.)
598
C H E RN F O R M S A N D H O M OT O P Y G RO U P S
Consider the case k < n . I t seems evident that f ( S k ) cannot cover all of sn 1'f k < n but this is actually false since we do not require our maps to be sm ooth ! Pean constructed a curve, a continuous map of the interval [0, 1 ] , whose image fi lled up entire square [0, 1 ] x [0, 1 ] ; see [H,Y, p. 1 23 ] . This map cannot be smooth, as you wil l show in Problem 22.3 ( 1 ). It is a fact that a continuous map of a sphere into an M n is homotopic (via approx_ imation) to a smooth one. Hence we may assume that f(Sk ) does not cover all of sn when k < n . Suppose then that the south pole of sn i s not covered. By pu shi ng away from the south pole we may push the entire image to the north pole; we have deformed the map into a constant map. Thus 7rk ( sn ) = 0 if k < n . Consider the case k = n . We know that homotopic maps of an n-sphere into itself have the same degree. A theorem of Heinz Hopf says in fact that maps ofany connected. closed, orientable n-manifold M n into an n-sphere sn are homotopic if and only if they have the same degree (the nontrivial proof can be found in [G, PD. Thus the homotopy classes of maps sn -+ S" are again characterized by an integer, the degree. Agai n, as for circles, one can construct a map of any integral degree. Thus we have, so far
�
7rk ( sn ) = 0 =Z
k
if 0
<
(22.24)
Hopf made the surprising discovery that there can be nontrivial maps of Sk onto sn when k > n > I ! We shall discuss one in Section 22.4.
22.3c. Exact Sequences of Groups A sequence of groups and homomorphisms
. . · -+ F � G � H -+ . . . is said to be exact at G provided that the kernel of g (the subgroup of G sent into the identity of H ) coincides with the image off, f ( F ) C G . In particular, we must have that the composition g 0 f : F -+ H is the trivial homomorphism sending all of F into the identity element of H. The (entire) sequence is exact if it is exact at each group. 0 will denote the group consisting of j ust the identity (if the groups are not abelian we usually use 1 instead of 0). Some examples. If
O� H� G is exact at H then ker h = i m f = O. Thus h is 1 : 1 . Since h is 1 : 1 , we may identify H with its image h ( H ) ; in other words we may consider H to be a subgroup of G. Ordinarily we do not label the homomorphism 0 � H ; we would write simply 0 -+ H � G . If
T H E H I G H E R H OM OTOPY GROUPS
is exact at G then im Iz H � G --+ o.
h
= ker g
=
11"dM)
599
G, and so h is onto. Again we would write
If
is exact, meaning exact at all the interior groups h is an isomorphism.
H, G , then h is I : I and onto; that is,
Consider an exact sequence of three nontrivial abeLian groups (a so-called short exact sequence) Then ker g is im f, which is considered the subgroup F = f ( F ) of G, and g maps G onto all of H . Note that if h = g ( g l ) and h g ( g 2 ) , then (g l - g 2 ) E f ( F) � F . Thus H m ay be considered as equivalence classes of elements of G, g2 "" g l iff g2 - g l is in the subgroup F. In other words, H is the coset space G / F ! (See Sections l 3 . 2c and 1 7.2a, but note that we are using additive notation for these abelian groups.) =
f
,,
f( F) = F �'
G
F I?
H = G/F
Figure 22.9
If the homomorphisms involved are understood, we frequently will omit them. For example, the exact sequence (2;[, is the group of even integers)
says that the even integers form a subgroup of the integers and ;[.2 sequence
�
;[,/2;[,. The exact
where the group of integers ;[. is considered as a subgroup of the additive reals, and w here lR --+ S I is the exponential homomorphism
r
E
lR
�
exp ( 2 n i r )
onto the unit circle in the complex plane (a group under multiplication of complex numbers) exhibits the circle as a coset space
600
C H E R N FORMS A N D HOMOTOPY G ROUPS
I n brief, a short exact sequence o f abelian groups is always o f the form o ---+ H
---+
G G ---+ - ---+ 0
(22.25 )
H
where the first homomorphism is inclusion and the second is projection. (As We saw in Section 1 3 .2c, G / H is always a group when G is abelian.) We have two examples from homology theory, as in Section 1 3 .2c 0 ---+ Zk ---+ Ck ---+ Bk - 1 ---+ 0 o ---+ Bk ---+ Zk ---+ Hk ---+ 0
(22 . 26)
a
See Problem 22.3(2).
22.3d. The Homotopy Sequence of a Bundle
For simplicity only, we shall consider a fiber bundle F ---+ P ---+ M with connected fiber and base. If F is not connected there is a change in only the last term of the following.
Theorem (22.27): If the fiber F is connected, we have the exact sequence of homotopy groups
The homomorphisms are defined as follows. Here we assume that the base point = * M of M is the projection Jr (* p ) of that of P , F is realized via an inclusion i : F ---+ P as the particular fiber that passes through * p , and * F = * p .
xo
i
F
P
F 1r M
Figure 22.1 0
It should be clear that a continuous map f : V ---+ M that sends base points into base points will induce a homomorphism f* : Jrk ( V ) ---+ Jrk (M), since a sphere that
THE HIGHER HOMOTOPY GROUPS
7rk(M)
601
g ets mapped into V can then be sent into M by f. This "explains" the homomor phi sms
and
induced by the inclusion i : F � P and the projection TC : P � M . We must explain th e remaining boundary homomorphism a : TCk (M) � TCk - 1 (F) . We illustrate the case k = 2. Consider f : 5 2 � M, defining an element of TC2 (M) . This is a map of a square [ 2 in to M such that the entire boundary j 2 is mapped to a base point Xo E M . all 4 faces are mapped by f to Xo
f
Figure 22. 1 1
This map can be considered as a homotopy of the map given by restricting f to the initial face [ defined by t2 = o.
f 1 2 --t-�-_
Figure 22.1 2
f restricted to this face is of course the constant map Xo . The base point * of P lies over Xo . By the covering homotopy theorem, f can be covered by a homotopy in P of the constant map [ I � * .
602
CHERN FORMS AND HOMOTOPY GROUPS
covering o f f(l2)
[IIJ Xo
Figure 22.1 3 Under f, the two sides and the bottom of the square are mapped constantly to Xo and the light vertical deformation curves are sent into closed curves on f (/ 2 ) since the top face is also sent to Xo . When these deformation curv�s are lifted into P from * they will become curves that start at * and end at points of the fiber Jr - 1 (xo ) = F holding *, but they needn' t be closed curves. Since the lines t l = 0 and t2 = 0 are mapped to x() , we see that these endpoints of the lifts of the deformation curves will form a closed curve in F, the image of some circle S l being mapped into F, that is, an element of Jr l ( F ) . This then is our assignment
Briefly speaking, the lift of a k-sphere in M yields a k-disc in P whose boundary is a (k - I ) -sphere in F. We shall not prove exactness, though some parts are easy. For example, consider the portion Jrk (F) � Jrk ( P ) � Jrk (M). A k-sphere mapped into F is of course also mapped into P . When this same sphere is projected down into M , the entire sphere is sent into a single point, and so is trivial . This shows that a sphere of P in the image of Jrk (F) � Jrk ( P ) must always be in the kernel of Jrk ( P ) � Jrk (M) . Conversely, if a sphere in P is in the kernel of Jrk ( P ) ---+ Jrk ( M ) , then its image sphere in M is contractible to the point X o . By covering homotopy, the original sphere in P can be deformed so as to lie entirely in the fiber F over Xo ; that is, it is in the image of Jrk (F) � Jrk ( P ) . This shows that the homotopy sequence is indeed exact at the group Jrk ( P ) . For proofs of exactness at the other groups (a few of which are easy) see [St] .
THE HIGHER HOMOTOPY GROUPS
7rk(M)
603
Note that at the last stage, 77: 1 ( P ) -+ 77: 1 (M) is onto because F has been assumed conn ected. A circle on M can be lifted to a curve in P whose endpoints lie in F and sinc e F is connected, these endpoints can be j oined in F to yield a closed curve in P that projects down to the original circle.
22.3e. The Relation between Homotopy and Homology Groups The homology groups Hk (Mil ; Z) deal with cycles (think of closed oriented k-dimen sional submanifolds of Mil ) ; a cycle is homologous to 0 if it bounds a (k + i )-chain. The homotopy groups 77:k ( M" ) deal with special cycles, namely k-spheres mapped into M il . A k-sphere is homotopic to 0 if it can be shrunk to a point, that is, if the sphere bounds the image of a (k + I ) -disk. This is the extension theorem (22. 1 7) . There are relations between these two groups . The following can be shown (but will not be used here). Let 77:1 be the fundamental group of a connected M. We know that 77: 1 is not always abelian. Let [77:1 , 77:l l be the subgroup of 77:1 generated by the commutators (elements of the form aba - 1 b- I ) . Then the quotient group 77: 1 / [77: 1 , 77:d turns out to be abelian and is isomorphic to the first homology group with integer coefficients 77: 1
--
[77: I , 77: d
;::::; HI ( Mil ; Z)
For the proof, see [G, H]. For the higher homotopy groups we have the Hurewicz theorem (Hurewicz was the inventor of these groups): Let M be simply connected, 77: 1 = 0, and let 77:j (M) , j > 1, be the first nonvanishing homotopy group. Then Hj ( M , Z) is the first nonvanishing homology group (for j > 0) and these two groups are isomorphic
The proof is difficult (see, e.g., [B, TD. As an example, we know that sn is simply connected for n > 1 . Also, we know that Hj (S" ; Z) is 0 for I :::: j < n, and Hn (S" ; Z) = Z (see ( 1 3 .23». Thus 77:j (sn ) = 0, for j < n and 77:,, ( sn ) = Z. Problems
22.3(1 ) Use Sard's theorem to show that if f : Vk � Mn is smooth and k < n, then f( V ) does not cover all of M. 22.3(2) Show that both sequences in (22 .26) are exact. (Note that the fi rst sequence
is defined only for k > 0, but if we define B_ 1 : = 0 the sequences make sense for all k ::: 0. ) Suppose we have a compact manifold and we consider the resulting finite simplicial complex, as in 1 3.2c. Suppose further that a field is used for coeffi cients. Then all the groups Ck , Zk , Bk , Hk are finite-di mensional vector spaces. Let C k , Zk , th , and bk be their respective dimensions ( recall that bk is the kth
604
CHERN FORMS AND HOMOTOPY GROUPS
Betti number) . For example C k i s simply the number o f k-simplexes in the corn . plex. ( Ck is independent of the field used, but we know that bk depen ds on the field .) Note then that, for example, bk d i m Hk = d i m ( Zk l Bk ) zk - fJk . =
=
(i) Show that Ck - bk
=
f3k + f3k- l
for all k ::': o. This is a Morse-type relation, as in Theorem ( 1 4.4 0) , w h ere now the Morse type number mk is replaced by Ck and qk is repl aced by fJk . We im mediately have C k ::': bk
that is, there are more k-simplexes than thekth Betti number. Fu rthermore as in the Morse inequalities, we have for an n-dimensional closed manifol d
n
n
L ( - 1 ) k Ck L ( - 1 ) k bk =
k =O
k =O
This is Poi ncare's theorem, expressing the Euler characteristic x ( M)
n
=
L ( - 1 ) k Ck k =O
as the alternating sum
=
(no. vertices) - (no. edges) +
. . .
of the Betti numbers. A special case of this was
noted i n Problem 1 6.2(1 ) .
(ii) What is t h e E u l e r characteristic o f
Sn , o f lR P'l, o f t h e Klein bottle?
(iii) Show that the Euler characteristic of a closed odd-dimensional ori e ntable
man ifold van ishes (H int: Problem 1 4.2(3» . Show that orientability is n ot really required by looking at the 2-sheeted orientable cover. 22.3(3) Let A c M be a subspace of M. Recall (from Section 1 4.3) the relative homol ogy groups Hp( M; A) constructed from relative cycles cp. A relative p-cycle c p is a chain on M whose boundary, if any, lies on A. Two relative cycles c and c' are homologous if c - c ' a mp+ l + a p , where m is a chain on M and a is a chain on A. The relative homology sequence for M mod A is =
. . . ->
Hp+ l ( M ; A)
a
->
Hp ( A)
->
Hp ( M)
->
Hp ( M; A)
a
->
Hp- ( A) 1
->
. . .
Here we are using the homomorphism induced by inclusion A -> M, the fact that any absolute cycle z on M is automatically a relative cycle, and the fact that the boundary of any relative cycle is a cycle of A (which bounds on M but not necessarily on A) . We claim that the relative homology sequence is exact. (i) Show that the composition of any two successive homomorphisms in the
sequence is trivial. (ii) Conclude the proof of exactness. (As an example, we show exactness at Hp( M) . From (i) we need only show that anything in the kernel of Hp( M) -+ Hp( M; A) must come from H p( A) . But if the absolute cycle z p of M is trivial as a relative cycle, we must have z a mp+ l + ap or z - a a m, which says that a is an absolute cycle on A and the abolute cycle z is homologous to it. Thus, as homology classes a -> Z and so z is in the image of th e =
homomorphism Hp( A)
->
=
Hp( M) . ) Simple pictures should be helpful.
S O M E C O M P U T A T I O N S O F H O M OTOPY G R O U PS
(iii) By considering the sphere Sn-1 C mology of S and of B, show that
Bn i n the n-ball, and knowi ng the ho
Hp ( Bn , Sn-1 ) =
What is the generator of
60S
Hn( B, S)?
0 for p < n Z for p = n
22.4. Some Computations of Homotopy Groups
How can one map a 3-sphere onto a 2-sphere in an essential way, that is, so that the map is not homotopic to a constant?
22.4a. Lifting Spheres from M into the Bundle P In the definition of a : Jrk (M) --+ Jrk - I (F) in Theorem (22.27), we have explicitly shown the following (the sketch for k = 2 works for all k ::: 1 ; one now lifts the image of the tk lines instead of the t2 lines).
Sphere Lifting Theorem (22.28) : Any map of a k-sphere into Mil (with base point xo) can be covered by a map of a k-disc into the bundlespace P, in which the boundary (k - i ) -sphere is mapped into the fiber F = Jr - I (xo). This has an important consequence for covering spaces. Recall that a covering space is simply a bundle over M with a discrete fiber.
Theorem (22.29): If Jr : M induced by projection is an isomorphism for k is 1 : 1 .
:::
--+
M is a covering space, then the homomorphism
2. Furthermore, for k
1
= 1
We first show that Jr* i s : 1 . Let f ( I k ) be a map of a sphere into M that when projected down is homotopic to the constant map to *. This homotopy can be covered by a homotopy of f into the fiber Jr - I (*) . But if k ::: 1 , the resulting map of a k-sphere into this fiber must be connected, and yet the fiber is discrete. It must be that the entire sphere is mapped to the single point *. Thus if Jr * f is trivial, then f itself is trivial, and Jr* is 1 : 1 for all k ::: 1 . We now show that Jr* is onto for k ::: 2. Let f(l k ) be a map of a k-sphere into M. This can be covered by a map ](l k ) of a k-disc into M whose boundary (k - I ) -sphere lies in the discrete fiber Jr - I (*) . If k ::: 2, this whole boundary must PROOF :
606
CHERN FORMS A N D HOMOTOPY GROUPS
collapse t o the point * . Thus J ( I k ) i s a map o f a k-sphere into M that projects via n to f ( I k ) , and n * is onto. 0 As simple corollaries we have nk (JR pn ) = nk ( S" ) nk (T")
=
(22.30)
for all k ::: 2. In particular, every map of a k
>
1 sphere into a circle T I is contractible
nk (JR/! ) = 0 nk (Klein bottle) = nk ( T 2 ) = 0
to a point!
22.4b. SV(n) Again
In Corollary (22.22) and in Problem 22.2(3) we saw that S U (n) is both connected and simply connected, n I S U (n) = O. We now show that n2 S U (n ) From the fibering S U (n - 1 ) homotopy sequence
PROOF:
=
-J>
0
S U (n )
-J>
S 2/! - 1 we have the exact
For n ::: 3 this gives
o
-J>
n2 S U (n - 1 )
and so n2 S U (n ) = n2 SU ( n - 1 ) =
.
.
.
-J>
n2 S U ( n )
-J>
0
= n2 S U (2) = n2 S 3 = 0
0
In fact, E. Cartan has shown that every map of a 2-sphere into any Lie contractible to a point n2 G
=
0 for every Lie group
group
is
(22.3 1 )
In Problem 22.4( 1 ) you are asked to show that (22 . 3 2) and thus every map of a 3-sphere into SU(n), for n ::: 3, can be deformed to lie in an SU(2) subgroup ! 22.4c. The Hopf Map and Fibering
The starting point for Hurewicz's invention of the homotopy groups must have been related to Heinz Hopf's discovery of an essential map of S 3 onto S 2 , that is, a map
SOME COMPUTATIONS OF HOMOTOPY GROUPS
607
rr :
S 3 � S 2 that was not homotopic to a constant map. (We have seen in (22.30) that this cannot happen in the case of a 2-sphere mapped into a I -sphere.) With our machinery we can easily exhibit this map. We know that when the group S U (2) acts on its Lie algebra (or on the trace-free hermitian matrices) by the adjoint action, the res ulting action covers the rotation group S O (3) acting on JR3 . In particular, S U (2) acts transitively on the spheres S 2 centered at the origin of its Lie algebra JR 3 . The stability sub group of the hermitian matrix (j3 is immediately seen to be the subgroup
wh ich is simply a circle group S ' . Thus we have the fibration S ' � S U (2) � S 2 From the homotopy sequence
0 = Jr3 S '
� Jr3 S U (2) � Jr 3 S 2 � Jr2 S ' =
0
we see that Jr3 S 2 = Jr3 S U (2) = Jr3 S 3 = Z, that is, Jr3 S 2 = Z and that the projection map
Jr : S U (2) � S 2 the Hopf map, is essential. We have shown that S 3 = S U (2) is a fiber bundle over S 2 with (nonintersecting) circles as fibers. This is the Hopf fibration
----- 5 2
Figure 22. 1 4 Here is another view o f the Hopf map. Consider the unit 3-sphere S 3
in ((: 2 . We then have a map Jr : S 3 � S 2 defined by (z o , z , ) � [z o , z t J , where the latter pair denote the homogeneous coordinates of a point in ((: p ' , that is, the Riemann sphere (see Section l 7 Ac). The inverse image of the point [zo , z t l consists of those
608
C H E R N F O R M S A N D H O M O T O P Y G R O U PS
multiples (AZo , AZ , ) in S3 , where A E ( C - 0) . Since 1 Zo 1 2 + 1 Z , 1 2 = 1 , we see that 1 A 1 2 = 1 , and so Jr - , [z o , z , ] consists of all multiples e i e of ( zo , z , ) . This is a ci rc le on S 3 passing through (z() , z , ) . Thus S 3 can be considered as the subbundle of the Hopf complex line bundle (of Section 1 7.4c) consisting of unit vectors through the ori gi n of 2 C , and then the Hopf map Jr : S 3 --+ S 2 is simply the restriction of the projecti o n map to this subbundle.
P roblems
22.4(1 ) Derive (22.32) .
22.4(2) We know 7r1 S O(3) = Z2 . Use S O(n) / S O( n - 1 ) that 7r1 S O(n) = Z 2 for n � 3.
=
sn -1 and inductio n to show
22.4(3) We have stated the exact homotopy sequence for a fiber bundle in the case that the fiber is connected. When the fiber is not connected (as i n the case of a covering) the only difference is that in the very last term , 7r* : 7r1 P -+ 7r1 M need not be onto, so that we do not necessari l y have that the sequence is exact at this last group 7r1 M. Accept this fact and go on to show that Theorem (22.29) is an immediate conseq uence of this exact sequence.
22.5. Chern Forms as Obstructions
Given a closed orientable submanifold
V 4 of M n , why is ( \ / 8 7r 2 ) Iv tree /\ e ) always an integer?
22.5a. The Chern Forms Cr for an SU(n) Bundle Revisited Let us rephrase some results that we have proved concerning Chern forms. First consider a U ( 1 ) bundle.
Theorem (22.33): Let E be a hermitian line bundle, with (pure imaginary) con
nection w ' and curvature e 2 , over a manifold Mil . Let V 2 be any closed oriented sUlface embedded in Mil. Then
� r e2 = r c , 2Jr Jv
Jv
is an integer and represents the sum of the indices of any section s : V 2 --+ E of the part of the line bundle over V 2 ; it is assumed that s has but afinite number of zeros on V. I t is only when this integer vanishes that one can possibly fi n d a nonvanishing section (that is, a frame over all of V). Next, instantons are associated with S U (2) bundles.
CHERN FORMS AS OBSTRUCTIONS
609
Theorem (22.34): The winding number of the instanton is given by 1 __2 � rr
j
�
tr Wu 1\ Wu
1\ Wu
=
- k.r C2
This represents the "number of times the frame eu on the boundary S 3 wraps around th e frame ev that is fiat at infinity." It is only when this integer vanishes that the fiat frame outside S 3 can be extended to the entire interior of S 3 . We have defined the Chern forms for a complex U (n) bundle E in Section 22. 1 c
(
det 1 +
ie 2 rr
)
=
1
+
=
1
+
C I (E) i
�
2rr
+ c2 ( E ) + . . .
(22.35)
1 8rr 2
tr e - - [(tr e ) 1\ (tr e ) - tree 1\ e ) ] + . . .
We have shown that each Cr is closed, dCr = 0, and thus defines a de Rham cohomology class, and that this cohomology class, with real coefficients, is independent of the connection used. The factor i is introduced to make each of the forms real (ie is hermitian). The factor I j2rr ensures that the "periods" of the Chern forms will be integers when evaluated on integral homology classes. We have already seen this in Theorem ( 1 7 .24) for the case of C I for a complex line bundle over a surface and have verified a very special case of this for C2 in Theorem (22.5). In this lecture we shall concentrate on the second Chern class C2 but for a general S U (k) bundle over a manifold.
22.Sb. C2 as an "Obstruction Cocycle"
Let Ck -+ E -+ Mn be complex vector bundle with connection. We shall be concerned with the case of most interest in physics, in which the structure group is the special unitary group G = SU C k). We are going to evaluate
where Z 4 is a 4-cycle on Mn with integer coefficients. For simplicity we shall in fact assume that z is represented by a closed oriented 4-dimensional submanifold of Mn . Let us consider the problem of constructing aframe of k linearly independent sections of the bundle Ejust over the cycle z. Since SU (k) is the structure group, this is equivalent to constructing a section of the principal SUCk) bundle P associated to the part of E over z. Each fiber is then a copy of G = SU (k) . We shall attempt to find a continuous section, since it can be approximated then by a differentiable one. Triangulate Z 4 into simplexes L). 4 , each of which is so small that the part of the bundle over it is trivial, rr - I L). � L). x G. We picture sections as frames of vectors.
610
C H E RN FORMS A N D HOMOTOPY G ROUPS
SU(k)
---+( ,",4 +)---
Figure 22.1 5
Now begin to construct a cross section . Over each O-simplex (vertex) �o we pick arbitrarily a point in Jr - I ( � o ) . Thus we have constructed a section of the bundle P over the "O-skeleton," that is, the union of all O-simplexes. Given � o , look at a 4-simplex � 4 holding this vertex. The part of the bundle over � 4 is trivial, Jr - I (�4) � � 4 X G . To construct a section over �4 is simply to give a continuous map f : � 4 --* � 4 X G of the form x � (x , g(x)) that extends the given f over the O-skeleton . Let � 1 be a I -simplex of the triangulation. This is a map a of I into Z 4 . Pi ck a � 4 holding � I . g is defined on the two vertices P and Q of � 1 = I ; that is, g(P) and g ( Q) are two points in G .
g ( Q) S U( k )
g (p)
f P
------'--""--
g
Q
Figure 22. 1 6
Since G = S U (k) is connected, these two points can b e joined b y a curve g : I --* G . Then define f : � I --* � I X G by f (t) = ( a (t ) , g et) ) . In this way we have extended
the cross section to each � 1 and thus over the entire I -skeleton. We now have the section f defined on the boundary of each 2-simplex �2; can we extend to the entire � 2 ? Letting Jr G be the local projection of Jr - I �4 = � 4 X G onto G, we see that JrG 0 f is a map of a � 2 ' topologically a circle, into the group S U (k) . We know from the "extension
C H E RN FORMS A S OBSTRUCTIONS
611
theorem" (22. 1 7) that this map can b e extended t o a map o f the "disc" � 2 i f and only i f it is homotopic to a constant map. B u t S U (k) is simply connected (Problem 22.2(3», and so any map of a circle into G is homotopic to a constant map, and Jie 0 / can be extended to a map F : � 2 -+ G . Define then a section over � 2 by l ex) = (x , F (x » , an ex tension of / over a � 2 . We have extended/to the entire 2-skeleton a/the 4-manifald z. SU(k )
f
Figu re 22. 1 7
We have defined / on the boundary of each 3-simplex. S ince each � 3 is topologically a 3-disc with boundary a topological 2-sphere � 2 , and fc = Jie 0 / is a map of a � 3 into G , we know that this map can be extended to all of � 3 if and only if /e : a �3 -+ G is homotopic to a constant. But Ji2 ( S U (k» = 0, (22.3 1 ), and thus fc is homotopic to a constant. As before, this allows us to extend the section / to the entire 3-skeleton. / is now defined on the 3-sphere boundary a �4 of each 4-simplex �4. But now Ji3 (SU (k))) = Z, (22 .32), and fc : a �4 -+ S U (k) need not be homotopic to a constant. We have met with a possible obstruction to extending / to the entire �4 in question ! We "measure" this obstruction as follows: The homotopy class of /e : a �4 -+ S U (k) is characterized, from (22.32), by an integer (call it j (�4)' and we assign this integer ta the 4-simplex �4. There is now a slight complication. Different � 4 'S in the 4-manifold Z4 will yield different trivializations of the bundle; that is, the S U (k) coordinate in the frame bundle over �4 changes with the simplex �4 . Consider for example, the case of S U (2) , which is topologically S3 . When we map a �4 into S U (2) we shall be using different copies of S U (2) , that is, different 3-spheres over different simplexes. If we change the orientation of the 3-sphere, our integer j will change sign. We shall assume that the fibers SU (2) can be coherently "oriented." Similarly, we shall assume that the fibers S U (k) can be coherently "oriented" so that the sign ambiguity in Ji3 S U (k) is not present. Steenrod called such a bundle orientable. In this manner we assign to each 4-simplex in the triangulation of the 4-manifold z a definite integer; thus we have a singular 4-chain on z with integer coefficients, called the obstruction cocycle. The reader should note that we did not really use the fact that the fiber F was S U (k) ; the only in/ormation that was used was that the fiber was connected and that Jij ( F ) = O/or j = 1 , 2, and that Ji3 (F) = Z.
612
C H E R N FORMS A N D H O M OTOPY G ROUPS
If each coefficient is ° it is clear that one can extend the section to the interior of each � 4 , and in this case we have succeeded in finding a section on all of z ! If, On the other hand, some of the i ntegers are not 0, it still may be possible to start anew and succeed. This will be the case if the sum 2: j ( � ) = 0, where the � are oriented so that Z 4 = 2: � . This can be shown using the cohomology theory of obstructions. We wish
rather, to show how this sum can be expressed as an integral involving curvature.
'
22.5c. The Meaning of the Integer j(�4)
The fact that rr3 (SU (k» = Z, proved in (22.32), is a result of two things. First, each S U Ck), k ::::: 3 , has the 3-sphere SU (2) as a subgroup and then the homotopy sequence shows that this 3-sphere is a generator for the third homotopy group of S U (k) . In other words, every map of a 3-sphere into S U (k) can be deformed so that its image lies on the S U (2) subgroup ! But then a map fc : ( a � 4 = S 3 ) --* S 3 has a degree, and this integer is j (�4) '
22.5d. Chern's Integral The partial cross section I : a �4 --* P on the simplex �4 is defined only on its boundary, but we can immediately extend it to all of � 4 with a small 4-ball B€ about its barycenter Xo removed; we merely make the SU(k) coordinates Ie constant along radial lines leading out to a �4 .
Figure 22. 1 8
We shall now compute the integral o f the second Chern form over � 4 ; since C2 i s a smooth form and is independent of the section
r
l':;4
r
E �O 1':;4 - 8, C2
C2 = lim
We will be brief since the procedure is similar to that in Section 1 7.3b. Let 2: 4 = I ( � 4 BE ) be the "graph" of the local section. Then -
r C = r C = r rr * C 1':;. - 8, 2 l" 2:4 2 l2:4 2 Now
1
C2 = - -2 [tr e !\ tr e - tree 8rr 1 = tree !\ e ) 8 rr 2 -
!\
e)]
613
C H E RN FORMS A S OBSTRUCTIONS
Let e u be a frame on the open U C Mn holding �4 for which n - 1 (U) is a product U x SU C k) . (J = (Ju is the local curvature 2-form for the vector bundle E. C2 = ( l j8n 2 ) tr (Ju 1\ (Ju . Then at a frame f = eu g we have
and from ( 1 8.2 1 )
n * c2
=
1 tr (J* 8n 2
-
1\
(J *
where (J * is the globally defined curvature form on theframe bundle, (J * = dw * +w * I\w * , where again w * is globally defined. The same calculation that gave (22.4) shows
[
]
1 d __ tr w * 1\ dw * + � w * 1\ w * 1\ w * (22.36) 8n 2 3 Thus the pull-back of C 2 to the frame bundle is the differential of a globally defined 3-form, the Chern Simons fo rm Thus, for the graph 2:: 4 of our section f over Z4 - U B" � r n * = -1tr w * 1\ dw * + w * 1\ w * 1\ w * (22.37) J2:: c2 � 2 a 2:: 3 n * c2
-
=
.
1
[
]
Figure 22. 1 9 Recall that w e have removed 4-balls from the 4-cycle Z 4 . The boundary o f 2:: over the 4-cyc\e Z4 consists of the part of the section f over the union of the boundary of the E -balls, but with orientation opposite to that of the balls (since f is 1 : 1 , 2:: carries an orientation induced from that of z).
r n * c2
J2::
Now over
=
[
1_ L Jr tr w* 8n 2 f(aB,)
_ _
U , for points of 2::
1\
dw * + � w * 3
1\
w*
1\
w*
]
(22.38)
(22.39)
614
CHERN FORMS AND HOMOTOPY GROUPS
where the section f is given b y f (x ) = e u (x) g (x ) . The triple integral for that BE in U will involve terms containing g - l dg and Wu =
(w} p (x )dx j )
In the integral (22.38), gather together all those terms that do not involve any dx ; one finds easily that the contribution of these terms is
1 _2 2: j
_
f ( a B, )
24n
tr g- l dg l\ g - l dg l\ g - l dg
(22.40)
A s in (22.3), w e see that the integral in (22.38) over f ( B BE ) represents the num ber of times that the image f (B BE ) of the E -sphere wraps around the S U (2) subgroup of S U (k) ! Furthermore, since the integrand, the Cartan 3-form [2 3 , is closed , Stoke s's theorem tells us that this also represents the number of times that the image of B � 4 wraps around S U (2) where � 4 is the simplex holding the given singularity. But this is precisely the index j ( � 4) that occurred in the obstruction cocycle in 22.5c. We have shown that
i- UB, C2
2: j (�4 ) + integrals involving dx (22. 41 ) O. The left side tends t o the integral o f C2 over the entire 4- cycle =
Now w e can let E ---+ z. We claim that the integrals involving dx all tend to O. Introduce coordinates X l , x 2 , x 3 , X 4 with origin at the singularity in question.
Figure 22.20
2
For B BE we choose the 3-sphere given by L X] = E . This can be parameter ized by angles a, e , and ¢> ; X l = E sin a sin e cos ¢> , x 2 = E sin a sin e sin ¢>, x 3 = E sin a cos ¢> , X 4 = E cos a . Each integral in (22.38), .1> (o B ) A , can be evaluated as the integral of a pull-back Ja B j * A . Let ¢> l = a, ¢> = e , ¢> 3 = ¢> . The pull-back of a term like g- l dg will be of the form G j (¢)d¢> j where the G j are differentiable and independent of E > 0 since we have extended the section to the interior of � 4 keeping g constant along radial lines ¢ = constant. Furthermore, since we have already taken care of the term involving [2 3 , each integral will also involve dx through w = j * w*, and dX i is of the form ( Bx i /B¢>i )d¢> j , which will have a factor of E . Since the functions wi p (x) are differentiable, we conclude that all the remaining integrals on the right-hand side of (22.39) vanish in the limit as E ---+ O. We have proved the following special case of a theorem of Chern.
2
615
CHERN FORMS AS OBSTRUCTIONS
Theorem (22.42): Let P be a principal S U (k) bundle over Mn . Then the integral
represents thefollowing. There always exists a section f : (Z4 U Pa ) -+ P over except, perhaps, for afinite number ofpoints { Pa }. About each Pa we construct a sma1l 3-sphere S; and map it into S U Ck) by means of the section f followed by the local projection Tee of the bundle into S U Ck). The image of Sa in S U Ck) can be deformed so as to lie on an S U (2) subgroup. Let ja (f) denote the number of times that the image covers S U (2), that is, -
z
j" (f)
:=
Brouwer degree of f 0 Tee : S;
-+
S U (2)
Then
Thus L ja (f) is independent of the section f! In particular, a section on all of z exists only if Jz C2 = O. It is also immediate that
1 C2 '-4
is an integer for each integer cycle z! Furthermore, this integral
� l tr e l\ e 8Te z.
is independent of the S U (k) connection used in the bundle! 22.Se. Concluding Remarks If our group G = S U (k) had not been simply connected, for example, if it had been U (k) , then, in our construction, we would have met an obstruction to a section of the k-frame bundle already at the 2-skeleton. The problem then would have been to try to construct a section over a 2-cycle, rather than a 4-cycle. The measure of the obstruction then would be the integral of the first Chern form c) over the 2-cycle. It turns out that for a U (k) bundle, the integral of C2 over a 4-cycle measures the obstruction to constructing not a k-frame, but rather a (k - I ) -frame section, that is, finding (k I ) linearly independent sections of the original bundle. It i s easy to see, however, that i f the group is S U Ck), then a (k - I ) -frame can then lead to a unique k-frame. For example, in ([ 2 , the most general unit vector orthogonal to ( 1 O) T is of the form (0 e i e f and is thus not unique, but if we demand that the pair ( 1 O) T and (0 e i e ) T have determinant + I then (0 e i e f must reduce to the unique (0 1 ) T . That is why we considered directly the search for a k-frame. The general situation is as follows. -
Chern's Theorem (22.43): Let E be a complex vector bundle with structure group U (k) and connection w over Mn. Then each Chern form Cr defines via de
Rham an integral cohomology class, that is, the 2r-form Cr has integral periods
616
CHERN FORMS AND HOMOTOPY GROUPS
on a basis of H2 r (M ; Z) . This class is called a characteristic cohomo logy c lass and represents an obstruction to the construction of a cross section to a bundle associated to E, namely the bundle of (k - r + 1 ) frames! Although the forms cr depend on the connection used in the bundle E, their periods do not. Note that we had considered orthonormal frames of p vectors in ]Rn in Problem 1 7 .2(3 ) ; the space of all such frames forms the (real) Stiefel manifold O (n )/ O (n p ) . Similarly, the space of all orthonormal frames of p complex vectors in en forms th e complex Stiefel manifold U (n) / U (n - p) . For example, the I -frames in en form the unit sphere and it is easily seen that s 2n - 1 is U (n) / U (11 1 ) , since U (n) acts transitively on this sphere. Besides Chern classes, there are other characteristic classes, the Stiefel-Whitney classes and Pontrj agin classes, which were defined before the Chern classes. We have dealt with the first and second Chern characteristic classes in terms of obstructions to constructing cross sections to U (n) bundles. For many purposes modern treatments con sider characteristic classes from a different, more axiomatic viewpoint. The interested reader might refer to [M, S] for such questions. _
S 21 - 1 ,
-
Problems
22.5(1 ) Consider the real unit tangent bundle To Mn to a compact orientable Rieman n ian n-manifold (see Section 2.2b). This fiber bundle has fiber Sn-l . Mimic our obstruction procedure to show that one can find a section on the (n 1 ) skeleton of a triangulation , and then one can find a section on all of Mn except perhaps for a fin ite collection of points. Hopf's theorem (1 6. 1 2) states that the index sum is the Euler characteristic. -
-
22.5(2) The unit normal bundle to a closed su rface v2 embedded in a Riemannian M5 is a 2-sphere bundle over v2 . Show that one can always find a section ; that is, there is always a unit normal vector field to a v2 in MS . What about for a v2 in an M4 ?
APPENDIX
A
Forms in Continuum Mechanics The above treatment of the problem of rotation may, in contradistinction to the usual method, be transposed, word for word, from three dimensional space to multi dimensional spaces. This is indeed irrelevant in practice. On the other hand, the fact that we have freed ourselves from the limitation to a definite dimensional number and that we have formulated physical laws in such a way that the dimensional number appears accidental in them, gives an assurance that we have succeeded fully in grasping them mathematically. Hermann Weyl, in Space-Time-Matter
A.a. The Classical Cauchy Stress Tensor and Equations of Motion
At a point p of a material body in � 3 we consider a small element of area da with unit normal n. The part of the body on the side of da to which n points exerts a force C, a "traction," on the part of the body on the other side of da . Under various assumptions, Cauchy shows ("Cauchy's first theorem") that this force is of the form (A. I ) or simply C = t (n da) . (The fact that t(n da) = -t( -n da) follows from looking at the equilibrium of a small thin disc.) The linear transformation (t i j ) is the Cauchy stress
tensor.
In the case of a perfect or nonviscous fluid, the force across the element da is normal to da, points in the direction -n, and has magnitude p da , where p is the pressure. In this case - p 8 ji
ti j =
(A.2)
A viscous fluid would also exert tangential stresses dependent on the velocity distribu tion. Consider a compact body B(t), which might be only a portion of a larger body in motion. In cartesian coordinates the equations of motion of B(t) are obtained from equating the time rate of change of momentum of B(t) with the total "body" force (for example, gravity) acting on B (t) and the traction force acting on the boundary aB (t) arising from the stress exerted on B(t) by the remainder of the body. Let
m3
:=
p
vol
be the mass 3-form. We assume conservation of mass
dldt r m 3 JB(t)
=
r c'f.v +3/at m 3 = 0
JB( t )
617
618
F O R M S IN C O N T I N U U M M E C H A N I C S
Let b be the external force density (per unit mass) ; w e have
djdt r v i m 3 = r h i m 3 + r t ij nj da laB(t) 1 B(t) 1B(t) = r j a x j vol3 h m3 + r l 1B Ct) E(t)
i
Thus, using
we have
[aVati vj ( ax}avi. ) ] m3 +
atij
=
hi m 3
+
( axt ) i}
a }
(A.3)
vol 3
As mentioned in the derivation of (4.46), this derivation makes no sense in a general Riemannian manifold nor in ]R3 with curvilinear coordinates, but the final formula, Cauchy's equations of motion, can be rewritten so that they make sense in these situa tions (A4) A "derivation" will be given in Section Af. We prefer to rewrite these equations using
Cartan 's machinery of exterior covariant differentials. A.h. Stresses in Terms of Exterior Forms
Equation (A.4) can be immediately extended to the case of an n -dimensional body in a Riemannian Mn , and this is the situation we shall eventually consider. All that we say, of course, will hold in the most important case, namely in ]R 3 with curvilinear coordinates. We shall use Cartan's calculus rather than vector or tensor calculus. One immediate reason for doing so is suggested by the boundary integral in (A 3 ) ; we could write it as
r
laB(r)
t ij n j da
=
r
laBCr)
(tCi ) n ) da ,
where tcn is, for each i, a vector with components t ij . We know that surface integrals should be written in terms of 2-forms, and when this is done, we know that the com ponents of the normal n and also the element of area da will play no role. Stokes's
theorem is not only independent of these metric concepts; it also has the same form in any curvilinear coordinates. This suggests that we redefine the stress tensor directly in terms of (n - 1 ) forms .
This is perfectly natural; the stress exerted on a hypersurface should be obtained by integrating an (n - 1 ) -form over the hypersurface. Since the result will be a vector rather than a scalar, the form should be vector-valued. The orientation of the hypersurface should play no role, but the stress vector for a tiny hypersurface element is reversed if
619
STRESSES IN TERMS OF EXTERIOR FORMS
we change "sides" of the hypersurface, that is, if the transverse orientation is reversed, and so we expect the stress form to be a pse udo form -
.
The Cauchy stress should be a vector-valued pse udo e n - I )-form t. -
When no confusion can arise, we shall omit the fact that t is a pse u do -form rather than a true form. In the language of vector bundles that we have developed in the third part of this book, t is a pseudo (n - I ) -form on Mil with values in the tangent bundle TM" . Locally we may write ,
(A.S)
Unless otherwise stated, e and cr will be dual frames for vectors and I -forms . Usually the frames will be orthonormal. For example, we may let the e basis be a global cartesian frame or perhaps the normalized version of the spherical coordinate frame a / ar, a / ae , and a/arP. In JR.n let e be cartesian. Then if B is a compact portion of a larger body we transver sally orient a B by the outward pointing normal and we interpret
{
JaB
t=
{
JaB
er 0 r = e r
{
JaB
r = er
{
JaB
r ldx
'
as the total traction that the part of the body outside a B exerts on B . We know that i n any Riemannian Mil , w e may write the (n I ) -form r , for fixed r, in terms of a vector t(r ) . Using (2.73) -
Thus (A.6) and
1 . ., ( ' ( rt : = t r ) l = _ t� , t ' Jg �
and Cauchy'S stress tensor t r i . Note that there is only one term in the expression for t r i , for r and i fixed. Look again at Eq. (A.4) with any dual frames e and cr for a Riemannian Mn . We are assuming that t = er 0 r is an (n - 1 )-form section of the tangent bundle; thus from Eq. (9.3 1 ) we have
exhibit the relation between the stress form
Vt
= V eer 0 r ) = Ver 0" r
f
+ er
0
dr
= er 0 (d f +
oIs
A t7 )
(A.7)
= er 0 V' f
where
is the connection form matrix for the frame e on Mn . If, temporarily, we use i, j, . for M indices (both sets run in this case from 1 to n ) , we would write, as usual,
r, s,
.
.
w
. for bundle indices and
.
.
r w s =
w
r
JS
cr j
(A.8)
620
FORMS IN CONTINUUM MECHANICS
I f w e recall that d of a n ( n - I )-form is an n-form related to the divergence of the associated vector field - see Eq. (4. 3 1 ) - we may rewrite (A.4) to read ( A.9)
which is our final form of Cauchy's equations. The reader might wish to see, at this ti me a specific computation involving Eq. (A.9) in spherical coordinates, given in Secti o A.g., part ( 1 ).
�
A.c. Symmetry of Cauchy's Stress Tensor in ]Rn
Consider again a compact body B (t ) moving in ]R1l , but use Cartesian coo rdin ates x and take for er the orthonormal a / ax r . Then w = O. We write equation (A.9) as
m n = b Q9 mil + d t (A. l O) dt where d / dt is the total or convective derivative operator. We wish to discuss the angular - Q9
dv
momentum of the body. Except in ]R 3 , angular momentum, like angular velocity, is not a vector but rather a skew symmetric second-rank tensor. A I -parameter group of rotations in ]R" need not have a I -dimensional axi s of rotation; look at the rotations in ]R2 ! In fact there is no I -dimensional axis of rotation in any ]Rn when n is even. We know that generators of rotations in any ]R" are given by skew symmetric matrices; see Section 1 5 .3c. We have used extensively the exterior product of two covectors, but now we shall deal with the exterior product of vectors, thinking of a vector v as a linear functional on covectors; v (a ) : = a (v) . Of course there is no essential difference here since we are dealing with orthonormal bases. As usual, er 1\ e., = e r Q9 es - es Q9 er and then, for any vectors v and w v
1\ W
=
1 2
- er
1\
es ( v r w s - vS w r )
with components forming a skew symmetric matrix, that is, a generator of rotations. In this case the plane spanned by v and w is rotated in the sense from w to v, the orthogonal complement is left-fixed, and the angular speed of rotation is the area of the parallelogram spanned by v and w. This replaces the notion of vector product v x w, which makes sense only in three dimensions. Of course the most general generator is not of the form v 1\ w but it is a linear combination of such generators. From the origin of ]R" one may form the position vector r = er x r . The angular momentum of the body B (t) , in Cartesian coordinates, is the generator obtained by integrating -r 1\ v Q9 m il
-H =
r er �2 JB(I) 1
= - er 2
1\
es
1\
j'
es (x r vs - x S v r ) Q9 m il
8(1)
(x r VS
-
x S v r ) Q9 m n
621
S Y M M E T R Y O F C A U C H Y'S S T R E S S T E N S O R
In the language of Section 1 8 .3, 1 /2
r er /\ es (xr v s - xS v ) 0 mil is an �� (n)-valued
n-form. Then, from Cauchy's equations (A. l O) _
dH
=
r
dt 1
B (t)
[ ( dt ) /\ v + r /\ ( dt ) ] dr
dV
o
m il
=
r r /\
1
B (t )
( dt ) dV
o
mil
r r /\ [b 0 m il + dt] lB (r ) Warning. d/dt represents the usual derivative, not an exterior derivative. Although we are dealing with exterior products, such as r /\ v, these are not forms, but rather =
"bivectors." In particular, we use the usual Leibniz rule for differentiating a product, not rule (3) of Theorem (2.53). We shall assume that for the material in question d H / dt must equal the i mpressed ques on B (t) tor
_ and so
Thus
r
1
B (t )
r
dH
=
dt
1
/\ d t =
r r /\ b 0 m" +
1
BU)
r /\ t = r d (r /\ t)
JB(r)
1
r
lB U )
B Ct)
dr /\ t
=
1
JB(t)
=
r /\ t
r
1
B (t )
d r /\ t + r /\ dt
0
and since this is true for each portion of B
(A. I 1 )
dr /\ t = 0 In components, omitting the tensor product signs as is usual I
0 = dr /\ t = erdxr /\ es t; = "2 er
/\
es [dxr /\ l'
.
-
dx' /\ tr ]
since er /\ es is skew in r, s . We conclude that (A. I I ) becomes
dxr
/\
l'
=
dx'
/\
t:
is symmetric in r and s
(A. 1 2)
What does this say about Cauchy's stress tensor t ij ? First note that
t1 dx r /\ dx J
and so
t!t E , J dx '
=
/\
s dxr /\ e = dx /\ t:
. . /\ dx" = .
=
tl dxs /\ dx J
!1E·, J dx ' /\ . . . /\ dx "
From (A.6) we conclude
(A. 1 3) which is Cauchy 's symmetry of the stress tensor in ]R" . Since the Cauchy stress tensor is a tensor, we conclude that the symmetry in (A. 1 3) holds in any coordinates, and likewise for (A. 1 2). We postulate then that (A. I 2) and (A. 1 3) hold in any coordinates on any Riemannian Mil .
622
FORMS IN CONTINUUM MECHANICS
A.d. The Piola-Kirchhoff Stress Tensors
Consider a reference body in Mn and a deformation (diffeomorphism) of this bod Let B be a compact portion of the original body with i mage (B ) . We Use loc coordinates X R for B and local coordinates x r for (B), and write in the form x r = x r (X) , using the notation of Section 2.7b. (Now, however, the coordinate systems X and x need not be related.) The Cauchy stress form t is a vector valued (n - I )-form on (B). We define the first Piola-Kirchhoff stress form on B, an (n - I )-form at X on B whose va lues are tangent vectors to (B) at x = (X ) , by
�
T=
a a a ® T' = ® T�dx A : = ® * V s ax ax ' ax s
-
-
�
-
(A. 1 4 )
that is Thus we pull back the forms t' = tJ, < . . . < j,, _ , dx j , !\ . . . !\ dx j,,-, but we do nothing to the vector values. We illustrate this in three dimensions in Figure A. I . T is called a S(v, w)
:=
T(v, w) : = t (
q):; i T(v, w)
v
Fig ure A.1
two-point tensor because the A indices in (A. 1 4) are tensor indices for B, the s index is a tensor index for (B ) . We say that T is an (n - I )-form section of the vector bundle obtained from pulling back the tangent bundle T (B) to B . This procedure makes sense even when B and (B) lie in different manifolds. We shall have more to say about this soon, but we refer the reader to chapter I of [M, H] . Of course y s is of the form interior product of VOL = .JGdX 1 !\ . . . !\ dXn with some vector T' = a/ax R ® T' R , leading to the two-point second-order tensor ysR . Although the form T' is simply the pull-back of t' , the tensor ys R has a rather strange expression, called the Piola transform of t, r
TsR
=
det
.Jg ( a X R ) s r ) (� t a x .JG ax r
623
STORED ENERGY OF DEFORMATION
Finally we introduce another measure of stress, the second Piola-Kirchhoff stress form S a/a ® d A . It will be an (n - I )-form at of B whose values will then the vector be tangent vectors to B at X. If v is an (n - I )-tuple of vectors at S (v) will have the property that S (v) <1>;:- l T(v) ; see Figure A. I .
=
XC s� X
X X,
=
(A. I 5 ) where
aXC) aXC) ( sC : = ( --= ax" <1>*t;' axS T'
If we write S 8 (A. 1 2)
X
d A
A
S 8 = <1>*
[ ( ��: ) dXQ ]
A
<1>*
([ aa�: ) ]
= ( ��:) (��: ) <1>* ( xQ = (��:) ( aa�: ) <1>* ( x d
d h
if
A
if )
A
�)
shows that S has the same symmetry as t ; that i s , the second Piola-Kirchhoff stress
tensor is symmetric
(A. 1 6) It makes no sense to talk of the symmetry of r S since r is a tensor index on <1> ( B ) whereas S is a tensor index o n B .
A.e. Stored Energy of Deformation
Consider again, as in the previous section, a body B in Mn and a deformation Bo = <1> ( B ) . We suppose that the body Bo is in equilibrium with respect to external forces and the tractions on Bo given by the Cauchy stresses. (Recall that Bo may be only a portion of a larger body.) From (A.9.), with v 0, we have the equilibrium equations
a
=
b ® mn
+
Vt = O
(A. I 7 )
Consider now a variation o f <1>, that is, w e have a I -parameter family o f diffeomor phisms written 8) with ¢o <1>. Let B (8 ) be the resulting bodies;
x = ¢e(X),
x = x(X,
=
624
FORMS IN CONTINUUM MECHANICS
o f course S (O) = (S) . Such a variation i s called "virtual" ; i t need not arise fro m an actual motion of the body S(O) . The variation vector has components ax r ox r . - a t: £=0 If, during the deformation, no energy is dissipated by heat flux, then the rate at which energy is stored in the body during this specific virtual deformation by the body force and the stress traction on the boundary is given by .
_
[ ]
(A. l 8)
(Unlike Eq. (A.3), this makes intrinsic sense since the integrands are not vector-valued. ) Put
and look at the first integral and apply (A. I 7)
r
lB(o)
oxs b' ® m
=
-
r
lB(o)
oxs v e
Now note the following; oXs V is a scalar-valued form and we claim that d of this form can be computed using exterior covariant differentials instead
d (ox" e) = V (oxs ) 1\ t' + ox" VV'
(A. 1 9)
This follows since V t' = de + wS r 1\ {; and, since oXs is covariant,
Vox" = d (oxs ) - oxr wr "
0
Looking again at the first integral
1B(O) ox.\ V t' = 1rB(O) V (oxJ 1\ t' - 1rB(O) d (oxs V ) oxs t' = r V (OXs ) I\ V - l 1B(O) aB(O)
- r
Putting this in (A. I S) then yields
OU =
r
lB(D)
V (ox, ) 1\ t'
(A.20)
We shall now rewrite this fundamental equation. First note, putting
oXs j r : = (ox., ) I r
V (ox\. ) 1\ t' = oXs l r dx r 1\ t' from the symmetry (A. 1 2) .
oU =
1B(O) 1
=
l (OXsl r
+ oXr ls )dx r
�s - (oxs j r + oX r ls )dx r 1\ v 2
1\
t'
(A.2 1 )
625
STORED ENERGY OF DEFORMATION
(8xs! r +8xr!s )
We have encountered the symmetric tensor before. In Problem 20. 1 (2) then we have shown, in essence, that if is the flow generated by the vector field the Lie derivative of the metric tensor is
¢,
8x, (A.22)
Thus (A . 2 3
=
)
1 1 _(A.* ax rR ) dX R <1>* t;' a g) r ( -2 B as s aX /\
-
'1',
But
[<1> * :s (¢; g )Lc = [:s <1> *(¢; g) Lc [:s (¢, O <1» * gLc a a = -([ ( ¢, <1» * g ]RC - GRe} 2- E R C 2 8E R C as as where E R C is the Lagrange deformation tensor 1 /2 { (¢, <1» * ds2 -dS2 } R C for the map ¢, <1>, introduced in (2.69). Thus 8U h 8ERC dX R SC (A.24) Let us write this out in terms of the symmetric tensor S A B R dX A 8U iBr 8ERC dX R S;'dX � A iBr 8E R C S;'dX � dXn = r 8E R C S;'E R A dX l iB r 8E R C S C E R A (_1_ ) VOL" -JG iB =
0
=
=
0
0
/\
=
/\
=
�
=
=
/\
/\ . . . /\
d
Thus, as in (A.6)
(A.25)
626
FORMS IN CONTINUUM MECHANICS
is the rate a t which energy is stored in the body during this specific virtual defo rma _
tion.
The equations (A.24) and (A.25) are precise, using no approximations. We do not claim that the same amount of energy is stored during two deformations starti ng at the same initial state and ending at the same final state; we expect the result to depend on the specific family of deformations from the initial to the final state. When the resul t is independent of the path in the space of deformations, the material is called hyperel astic .
A.f. Hamilton 's Principle in Elasticity
Equations (A.4) and (A.9) are the basic equations of motion. They make sense in a Riemannian manifold, but our derivation of them was Euclidean and could not be carried out in a curved M . Frequently it is argued that these laws are local, and a "tiny" portion of a curved Riemannian M almost looks like Euclidean space, but it is not possible to convert such statements into a derivation. We have simply accepted them as postulates. The same is true of the symmetries of the Cauchy and second Piola Kirchhoff stress tensors. Our equations of rate of stored energy of deformation suffer the same fate. In this section we shall start anew with what I believe is a more fundamental view point, namely Hamilton's principle. This will replace the Newton-Euler principle, equating the total force on an extended body with the time rate of change of total momentum, since neither the total force nor the total momentum can be defined in curvilinear coordinates. We shall take as a basic property of an elastic body the fact that it takes work to deform the body, and we shall, assume that during a deformation the rate at which energy is stored in the body is proportional to the rate of change of strain. In other words, we shall use Eq. (A. 24) or (A. 25) to define the stress tensor. We proceed with this plan, which in its main thrust goes back to George Green in a paper of 1 837. We shall be very brief since we have carried out most of the calculations needed already. We consider again a body B , the "reference body," sitting in a Riemannian M n , as we did in Section A.d. Let B (O) = ( B ) be a deformation of the body. Let B (t ) =
o
J Tdt - J o Udt + J o Wdt
=
0
(A.26)
We do not write 0 J U dt because we do not assume that there is a stored energy function U: there is only the differential 0 U . All time integrals run from 0 to t l '
627
H A M I L T O N 'S P R I N C I P L E I N E L A S T I C I T Y
cI> * m M, T �2 lB(r I) /\ axat , axat ) m = �2 1Br v, v M is the kinetic energy of the body. Here V ( X, t , ax ( X, t , a t is the velocity field in B is a vector based at in the reference body, a vector field whose value at (X, t, the point x ( X, t, in B (t ) . r 8x,b'm + r 8x,r;dxa 8W = lB(I) laB(I) lBr 8x,B'M laBr 8x/i' where �f ' is the pull-back of the traction form r' to the reference boundary B . Using conservation o f mass,
=
=
<
8) : =
>
8)
8)
8) /
+
=
a
We assume that the rate at which energy i s stored i n the body during the variation is of the form
8U
E
=
1 8EA B SA B VOL
(A.27)
S
-
where is the Lagrange deformation tensor and is some symmetric tensor. From we may construct the associated (n I )-form (as in (A.6»
S
and, as in (A. I S) and (A. 1 4), the associated vector-valued forms T and t. Later an we shall see that t should be interpreted as the Cauchy stress form. We now look at the three integrals involved in (A.26). The second version of the integral for T involves the reference body B , which is not changing in time. We have varied ( V, V ) many times in our discussions of geodesics and mechanics. Since the variation vector vanishes at t = 0 and t = we easily get
8x
t,
8 J T dt - J dt 1 8x, [ Va� ' ] M
(A.2S)
=
Here the covariant derivative of V along the parameterized curve tangent is V, is, from (S .42),
- -- V J w'jk V represents the Christoffel connection at x ( X, t). V Vi at
-- =
a vi at
a v' . ax j = J at at
+ V k w'. k
+
.
.
x ( X, t), whose =
k
where w Our derivation o f (A.2S) from (A.20) can b e reversed to give, instead o f (A.27),
r V (8xs ) 8U lB(I) lB(r I) 8xs/,dx' lBr cI>�[8xs/,dx' t' ] ( 8x' = lBr 8xs/, 8X R ) d X R T s Now in this integral over the reference body B , ( 8xs ) is a field whose value at X is a covariant vector at x . (This is similar to having a vector field defined along the map cI>, a generali zation of the situation in Section 1 0. 1 .) We define the covariant derivative =
/\
e
/\
=
/\
t' =
/\
628
FORMS IN CONTINUUM MECHANICS
(OXs ) / R at X as follows: A s w e move along the XR coordinate curve with tangent 8/8XR the image curve in B (t ) has tangent (ax r / a X R ) 8/8x r and we define ax r (OXs ) / R : = (A 29) a XR
(ox,)/r ( )
Thus 0 U becomes
(A 30)
V (ox,) oXs/
X R is the covariant differential in the so-called pull-back bundle = where Rd to We can also think of the covariant differential for a two-point differential form, such as YS , in terms of Cartan's calculus. The exterior covariant differential of should be of the form
B.
ys
where QS r = Q� r dXA is a suitable connection form matrix. For the pull-back bundle the connection is given in terms of the connection w on Mn by ax Q d XA Q S := cP* (WS ) cP* (WS dx Q ) WS (A.3 1 ) a XA
r
.
r
=
ar
=
Qr ( ) __
Then, using (A 1 9), ( A 3 0 ) becomes
Thus
OU =
1a B oXs T"' - irB oXs V TS
- J o Udt J dt i oXsVT"' - J dt 1B Xs T o
=
(A32)
'
Hamilton's principle (A.26) then becomes
- J dt i oXr ( Va� r ) M + J dt i oxsVTs - J dt 1B oXs Ts J dt {i oxr B r M + 1B OXr:f r } 0 J dt is oXr [{ - Va� r B r } M V T r ] + J dt 1 oxr [:f r - T '] +
or
=
+
+
B
=
0
J dt is oXr [{ - Va�r B r } VOL + VT r ] (A.33) r + J dt 1 oxr [:r - T '] 0 B By taking variations ox that vanish on and near aB we get Cauchy's equation in the reference system vv r } VOL VT r + B r PB VOL (A. 34) { at
Thus,
+
PB
=
PB
=
629
SOME TYPICAL COMPUTATIONS USING FORMS
{
k} -
I n terms o f the second-rank Piola tensor, w e may write this as
PB
a vr j at + V wjr k V
where Tft = 1 /.jC [a/aXA (.jC FA ) ]
w r as (oxa / o X A ) .
_
+
T/rA A
+
PB B r
(A.35)
Q�s T'A and, from (A.3 l ), Q�s
=
We now have J d t Ja B oxr [:f r - F ] = O . Considering variations o n oB then shows
:f r
=
Tr
on
(A.36)
aB
which are boundary conditions that must be satisfied. This last equation can also be written = t'
rr
on
a B (t)
which, finally, allows us to identify t with the Cauchy stress form.
Equations (A.32) and (A.33), with given W = W (X), are useful for discussing the motion of the body. Consider a body in 1R 3 with Cartesian coordinates X in the reference body B and x in B (t ) . The goal is to find xr = xr (x, t). One relates the second Piola stress tensor S to the Lagrange deformation tensor E by means of a generalized Hooke's law, SA = a function of E RS . For example, in a linearized theory one might write
SuA v
=
for some constant coefficients C C � s . Now in Cartesian coordinates
EA B =
C uA RS v E RS
� [ (:;:) ( :;: )
- OA B
]
and so SC v and then Tt v = (oxa / o XA ) SC v become complicated functions of ox/o X . Finally, (A.34) becomes
P B (X) =
[
o2xr (x ' t)
o t2
d[T;< B d XA
/\
] dX '
/\
dX 2 /\ d X 3
dX B ] + P B (X) B r d X '
/\
a complicated partial differential equation for x = x (X, t).
dX 2
/\
dX 3
A.g. Some Typical Computations Using Forms
The supreme misfortune is when theory outstrips performance. Leonardo da Vinci
The use of differential forms can reduce the complexity of the computations of continuum mechanics somewhat, especially when curvilinear coordinates are involved. Consider, for example, spherical coordinates in 1R3 . There are 1 8 Christoffel symbols to compute in the tensor formulation based on the usual coordinate bases, whereas if we use Cartan's method and orthonormal frames there are only 3 connection forms to determine ! This is the same philosophy as used in Section 9.5c.
630
FORMS IN CONTINUUM MECHANICS
This section has been contributed by m y engineering colleague Hidenori Murakam i. The first two parts deal with the equilibrium equations and the rate of strain tensor respectively, when dealing with spherical coordinates, while the last part di scuss e stress rates in any coordinates. ( 1 ) The equilibrium equations in sphericaL coordinates. The metric is
�
with respect to the coordinate basis [ a / a r, a / a e a / a ¢] To get an orthonorm al bas is it is immediately suggested that we define a new basis of I -forms by ,
a =
[ :: 1 [ a'"
with dual-vector basis
e = [ er ell e",] = A vector v has two sets of components
v = vr
a ar
+
V II
a ae
.
�
r e
r sin ed¢
1
I __ � ] [�ar ' �� ' r ae r sin e ¢
+ v'"
a
a a¢
=
vrer + v li ell + v ¢ e¢
where a boldfaced index denotes that the vector component is with respect to an orthonormal frame, that is, a physical component. Thus
v¢
=
r sin e v'"
Likewise, forms will have both coordinate and physical components. For example, one part of the Cauchy stress form is
t = eli ® tre dr /\ de = eli ® ttlidr /\ (rde)
and so fL rli = r - l fl-r e . We would like to stress one point. Most engineering texts deal with the components of tensors, and components are almost always the physical ones. We prefer to carry out computations with the forms themselves rather than the components, for example, tre dr /\ de rather than j ust trll For an electrical example (Problem 5(3)), the magnetic field due to a current j in an infinite straight wire is the I -form * �g = 2jd¢ in cylindrical coordinates. The derivation of this, using forms, was a triviality. Furthermore, an ex pression such as this is immediately available for yielding numbers by integration. The I -form has physical significance even if the coordinate component B", = 2j does not. The physical component B ¢ = 2j / r is much better for indicating the 1 / r dependence of the field. Our philosophy then is initially to deal with the forms for derivations and computations, and then translate into physical components if necessary or to compare with the textbooks. We shall need the matrix of connection I -forms w for our orthonormal basis. Letting a / ax be the Cartesian basis, using x = r sin e cos ¢, y = r sin e sin ¢, and z = r cos e , li
li
SOME TYPICAL COMPUTATIONS USING FORMS
we get
er =
a = ar .
631
( ax ) a + ( ayar ) aya + ( ara z ) aza ar
ax
a = sm e cos ¢ ax
a ay
a az
+ sm e sin ¢ - + cos e -
.
ee and e¢ may be expressed in the same manner in terms of the Cartesian frame. We then have
a e = -P ax or [er e(l e¢l = [ ajax ajay ajazl P , where P is the orthogonal matrix P=
[
sin e cos ¢ sin e sin ¢ cos e
cos e cos ¢ cos e sin ¢ - sin e
- sin ¢ cos ¢
o
]
The flat connection r for the Cartesian frame ajax is r = O. Under the change of frame e = ajaxp we have the new connection matrix, as in (9.4 1 ) ,
[
yielding the skew symmetric matrix
W=
0
-de
de sin ed¢
- Sin ed¢ - cos ed¢
o
cos ed¢
o
]
(A.37)
We may now write down the equilibrium equations (A. 1 7). The Cauchy stress form is
t = er
18) r +
ee
18)
tt + e¢ 18) t1'
Each t , for i = r, e , or ¢ , is a 2-form, which can be expressed either in terms of physical components
or, as we prefer, in terms of coordinate components
Of course
4 e = rt�e 4 ¢ = r sin e 4¢
fi'11 ¢ =
In any case we have
r 2 sm e tiO ¢ .
(A.38)
632
FORMS IN CONTINUUM MECHANICS
Thus
[Vf] Vt&vt4> [df] dt&diJ' [ ed¢ d¢ ddf -dd t!!. dr d¢d¢- d¢ tt dr d [( ar 4> ) e 1 dr d dA. r2 dr d d¢, [( � ) 1 r2 r2 r -2 -(r2t�",) -(r r-I(&,,,, =
0 de
+
-de o
sin
cos e
Look, for example, at the equation for V e' .
ve = f
-
=
e /\
e /\
to - sin e
Since pb r vo 1 3 = p b r ar
ar
/\
sin e
-
+
a t: af � + � ae a¢
Putting in (A.38) and dividing by components a
/\
e
a¢ /\
/\
e /\
'f'
we get the first equilibrium equation
+ tfr 4>
'" - sin e trli
+ pbr
sin e = 0
sin e yield the equilibrium equation in physical a
a�
ae
a¢
sin e ) - I - (sin e t�", ) + (r sin e ) - I �
ar
+
/\
o - sin e trli'" + trt/J
af
�
e
t4>
sin e
/\
ar af � - � ae
=
/\
-
tfo ) + pbr = 0
The equations for v t� and vt1' are handled in the same way. Note that these computations are straightforward and can be carried out without committing complicated formulas or expressions or methods to memory. (2) Consider the metric tensor
The rate of deformation tensor in spherical coordinates. ds2 = gij dx i
® dx j
in any Riemannian manifold. If v = (8/ 8x i ) V i is a vector field, then, as mentioned in Eq. (A.22), the Lie derivative of the metric, measuring how the flow generated by v deforms figures, is given by
where
c�v(gij dXi
® dx j )
= 2dij dx i
® dx j
defines the rate of deformation tensor, which plays an important role when discussing the linearized equations of elasticity involving small displacements. We shall now compute this tensor in spherical coordinates, not by using covariant derivatives but rather by looking directly at the Lie derivative of the metric tensor
. dx1). a ¢;(gij dx'. dx1).
cS:v (gij dx '
®
=
O
e
®
633
SOME TYPICAL COMPUTATIONS USING FORMS
¢e exterior
where is the flow generated by v. We shall do this by using only the simplest properties of the Lie derivative. We have mainly discussed the Lie derivative of vector fields and forms, where Cartan's formula (4.23) played an important role. Since we are dealing now with quadratic (symmetric) forms, (4.23) cannot be used here. However, we still have a product rule
and the basic
!l:v( f ) = v ( f) = df (v) for any function
f . Also
Thus (A.39)
.f.v (gij dxi 0 dxj) in spherical coordinates. Put ar dr al.l rde and a ¢ = r sin ed¢ S'v(dr 0 dr + rde 0 rde + r sin ed¢ 0 r sin ed¢) = [(S'vdr) 0 dr dr 0 (.\:'vdr)] W''v rde) 0 rde + rde 0 (S'vrde)] + [(5:'vr sined¢) 0 r sin e d¢ + r sined¢ 0 (S'vr sined¢)] Look for example at the term Cf.v rde) 0 rde in the second bracket. Since We are now ready to compute
=
=
+
+
( A . 40
we have
+ r2 [( aa�) dr + ( aa� )de + ( �� ) d¢] 0 de [rvr r 2 ( aa�) ] de 0 de + r 2 [ ( aaV; ) dr 0 de + (��) d¢ 0 de ] [ ( a �) ] 0 a e r [ ( aaV; ) a r 0 a e ] = � a + (sin e ) - I ( �� ) a ¢ 0 a l.l +
+
a l.l
+
)
634
FORMS IN CONTINUUM MECHANICS
[ r -r ( ae ) l e ( ) r
e r-ar ( -r ) r (Je
Introducing the physical components of v this last expression becomes
vr
- +
a vO -
1
1 sin e
_
rde(J vrd
+ _
a
(J 0 (J +
a vO a¢
vO
(J 0 (J e
¢> (J 0
The term 0 C'f e ) in the same bracket will yield the same result but with (J i 0 (J j j replaced by 0 (J i . We see that the total contribution of the second bracket term in (A.40) is
r ( ) .'fv (gudxi dxj ) 2dudxi dxj d The Lie derivative of the Cauchy stress form. tU , t , tu , l dx1. coordinate +
-!-
{ (J ¢> 0 (J II + (J e 0 (J ¢> }
a vO a¢
sm e
and the remaining brackets are done similarly. I f w e put 0
0
=
i = 2 jjO" 0 (J j
one can then read off the physical components dro and so on. (3) The Lie derivative of the Cauchy stress tensor arises in continuum mechanics in various forms. In [M, H, p. 1 00] the Cauchy stress is considered in its various two-index versions ; or with the volume form attached. In the spirit of this Appendix the natural candidate is the Lie derivative of the vector-valued 2-form ei 0 e To make comparisons with [ M , H] we shall use a general basis ei = 3 a/ax i in JR . We define the Lie derivative of the vector-valued form ei 0 e with respect to the time-dependent vector field v by
1
[
.�vH/at (ej 0 e ) : = .�vH/at (ei ) 0 e + ei 0 £v+% t ( e ) . . ae = v e; ] 0 t + ei 0 at + £v ( t )
[,
For the bracket term [v, ei ] 0 ti we have, from Eg. (4.6), . . a a vj [v, ei ] 0 t = - [ei ' v ] 0 t = v 0t = ' ax i a i Also,
a t! 1 at + ..I:\ ( t' ) = 2:
[ ] . ( )j t x { ( -tr- ) dxJ. dxk + £v[ tj. k dxJ. dxk } ae k
. - { ( _atJ_ ) dxJ. dxk kJ ( -axr ) dxr dxk
e 0
/\
/\
Using (A.39) again we get for this expression 1
2
a t! k .
+t
/\
a vj
J. k )dxJ. dxk tJk ( -axr ) dxJ dxr } /\
+ v et
/\
+
.
a vk
.
/\
]
.
635
CONCLUDING REMARKS
[ at!.at vr ( at!axr )] dxi at! a vr (a t! axr), at! )] . e [ at! vr ( i at ax r
The first two terms here yield the convective derivative 1 -
2
which we may write as
--....l!... +
/ t+
--....l!...
/
/\
dxk
and our final result is
$.'v+a/at(ei 0 n = 0 + " [Ii ( a vri ) + Vii ( a vr ) - ( a vi ) II./ir k ] dx i /\ dx k + ei 0 L.- 'tk J< k ax r ax k ax r -
(A4 1 )
A.h. Concluding Remarks
Since the first printing of this book I have learned that Elie Cartan, in his generalization of Einstein's theory of gravitation, introduced in 1 923 a vector valued 3-form version of the stress tensor in space-time M 4 and combined this with a vector valued 3-form version of the energy momentum tensor Cartan's version of the symmetry of these tensors is exactly as in our equation (A. 1 2). For these matters and more see the translation of Cartan's papers in the book [Cal , and especially A Trautman's Foreword to that book. Finally I should remark that L. Brillouin introduces the three index version of the stress tensor in ]R 3 in his book [Br, p. 28 l f f ] . In particular, Brillouin writes down the symmetry = in the form K = 0 ; this follows easily from our version (A I 2) since both sides of this equation are n-forms. He abandons the three index version since the two index version seems simpler. He made this judgment, I believe, because he, like most scientists other than Cartan, dealt with the components of forms, rather than with the forms J themselves, and made no use of Cartan's methods. I believe that it is a distinct advantage to remove the apparent dependence on unit normals and area elements, especially since deformations hardly ever preserve normals or areas of surface elements. The readers of this book are now invited to make their own judgments.
pui u k .
t il t i i
I:i t i i
t! dxJ
t! J
APPENDIX
B
Harmonic Chains and Kirchhoff' s Circuit Laws
Chapter 14 deals with harmonic forms on a manifold. This involves analysis in infinite dimensional function spaces. In particular, the proof of Hodge's theorem ( 1 4.28) is far too difficult to be presented there, and only brief statements are given. By considering finite chain complexes, as was done in section 1 3 .2b, one can prove using only elementary linear algebra. In the process, we shaH consider cohomology, which was only briefly mentioned in section 1 3 .4a. In the finite dimensional version, the differential operator d acting on differential forms is replaced by a "coboundary" operator 8 acting on "cochains," and the geometry of 8 is as appealing as that of the boundary operator a acting on chains ! As an application we shall consider the Kirchhoff laws in direct current electric circuits, first considered from this viewpoint by Weyl in the 1 920s. This geometric approach yields a unifying overview of some of the classical methods of Maxwell and Kirchhoff for dealing with circuits. Our present approach owes much to a paper of Eckmann [E] , to Bott's remarks in the first part of his expository paper [Bo 2], and to the book of Bamberg and Sternberg [B, S ] , where many applications to circuits are considered. We shaH avoid generality, going simply and directly to the ideas of Hodge and Kirchhoff.
afinite dimensional
analogue ofHodge's theorem
B .a. Chain Complexes
A (real , finite) chain complex C is a collection of real finite dimensional vector spaces { C p } , C - I = 0, and boundary linear transformations a = ap : Cp
---+
Cp _ 1
such that a 2 = a p _ 1 0 a p = o . Chapter 1 3 i s largely devoted to the (infinite dimensional) singular chain complex C (M ; R) on a manifold and the associated finite simplicial complex on a compact triangulated manifold. We shall illustrate most of the concepts with a chain complex on the 2-torus based on simplexes (as in Fig. 1 3 . 1 6) but rather
not
636
637
CHAIN COMPLEXES
on another set of basic chains illustrated in Figure B. l . This chain complex is chosen not for its intrinsic value but rather to better illustrate the concepts.
Vii
EI
-rl
G
E3
1'2
E3
1'2
E4
G
J
1
EI
Figure 8.1
The vector space Co is 2-dimensional with basis the vertices v , and V2 . C , is 4dimensional with basis consisting of the two circles E , and E4 and the two I -simplexes E2 and E3 , each carrying the indicated orientation. C2 has as basis the two oriented cylinders F, and F2 . We call these eight basis elements basic chains. i A general I -chain is a formal sum of the form c = L: a i Ei , where the a are real numbers. This means that c is a real valued function on the basis { Ei } with values C ( Ei ) = a i . Similarly for Co and C2 . For boundary operators we are led to define a = ao (Vi ) = 0
i
= 1, 2
a = a , E , = V , - V I = 0 , a 1 E2 = V2 - V I , aI E 3 = v , - V2 , a, E4 = V2 - V2 = 0 a = a2 FI = E I + E2 - E4 - E2 = E I - E4 , a2 F2 = E4 - E I i and extend a to the chain groups by linearity, a L: a Ei = L: a i a Ei • Using the usual column representations for the bases, E3 = [0, 0, 1 , O] T , etc . , we then have the matrices
ao
=
0
al =
[
0 0
-1 1
-1
�]
(B . I )
We may form the homology groups (vector spaces) of the chain complex. Hp (C) : = ker(ap ) /Im(ap+' ) , which are again cycles modulo boundaries. One sees easily that the bases of the homology vector spaces can be written
yielding the same bases as ( 1 3 .24) for the finite simplicial chains on the torus. There is no reason to expect, however, that other decompositions of the torus will yield the same homology as the simplicial chains. For example, we could consider a
638
HARMONIC CHAINS AND KIRCHHOFF'S CIRCUIT LAWS
new chain complex o n the torus where C2 has a single basic chain T , the torus itself while C I = 0 and Co is the I -dimensional space with basic O-chain a single verte v , and with all ap = O. The homology groups of this complex would be Ho = { v }, H I = 0, and H2 = { T } , which misses all the I -dimensional homology of the toru s. We have chosen our particular complex to better illustrate our next concept, the cochain s.
�
B .b. Cochains and Cohomology
A p-cochain a is a linear functional a : CI' -+ lR on the p-chains. (In the case when C p is infinite-dimensional one does require that f vanish except on a finite number of basic chains !). The p-cochains form a vector space C P : = CI' * , the dual space to Cp ' of the same dimension. Thus or I -forms. Cochains are not chains. However, after one has chosen a basis for p-chains (the basic chains), each chain is represented by a column = and a cochain, with respect to the dual basis, may be represented by a row a = . . aN ] . However, for our present purposes, some confusion will be avoided by by Then the value of the cochain a on the chain is the matrix product a = We may also think, in our finite dimensional case, of a chain as a function on cochains, using the same formula
not chains correspond to vectors while cochains correspond to covectors c [c l ,[a...I, cN. .V cochains al(cs)o acolT c.umns. c representing c(a): = a(c) a T c =
(B.2)
In our simple situation there will always be basic chains chosen so there is basically no difference between chains and cochains: both are linear functions of the basic chains, but just as we frequently want to distinguish between vectors and I -forms, so we shall sometimes wish to distinguish between chains and cochairls, especially in the case of Kirchhoff's laws. We define a coboundary operator 81' : CI' * -+ C 1'+ I * to be the usual pull back of I -forms under the boundary map ap+ 1 : Cp+ 1 -+ Cp o Ordinarily we would call this ap+ 1 *, but as we shall soon see, * is traditionally used for the closely related "adjoint" operator.
is defined by (B.3) or, briefly
(p
for each + 1) chain C. As usual the matrix for 81' is the transpose of the matrix for a 1'+ I , again operating on columns. It is immediately apparent that 82 = 8 0 8 = 0
(B .4)
639
TRANSPOSE AND ADJOINT
If 0 a P = 0 we say that a is a p-cocycle, and if a = 0{3 1'- ' then a is a coboundary. It is clear that every coboundary is a cocycle. In the case when C p is the infinite dimensional space of real singular chains on a manifold Mn , then an exterior p-form a defines a linear functional by integration (called I a in our discussion of de Rham's theorem) a( ) =
c
1
a
and so defines a cochain. Then Stokes 's theorem d a ( ) = a ( a ) shows that d A closed form defines a cocycle and an exact form a coboundary. The analogue of the de Rham group, �R P = closed p-forms modulo exact p-forms, is called the pth (real) cohomology group for the chain complex
c c
as a coboundary operator.
behaves (B.5)
T2
Consider the chain complex on pictured in Figure B . l . Consider the basic chains also as cochains; for example, I:� , is the l -cochain whose value on the chain is 1 and which vanishes on and Then o �� , ( F, ) = (;� , (a F, ) = i:� , E2 ) = + 1 , while similarly M" ( F2 ) = - 1 . Thus we can visualize o i:� , as the 2-chain F, F2 .
(E, E2 -E4E, E2 , E3 E4. -2 r as a chain, � a r E, (ar E, E , E, E4 2 E2 . -
-
M � , = F,
F
In words, to compute M � , we take the formal combination Fr of exactly those basic 2-chains { Fr } whose boundaries meet , a r chosen so that a Fr ) contains with coefficient 1 . Note that
since F, is the only basic 2-chain adjacent to
but a F, =
does not contain
These remarks about o &. , and 0&.2 also follow immediately from the matrices in (B . 1 ), putting 0 , = aT . Observe that 0&.4 = F2 - F" and so (B .6)
E, E4 E, E4 cannot bound.
The I -chain + is not only a cycle, it is a cocycle. We shall see in the next section that this implies that +
B .c. Transpose and Adj oint
finite column
We shall continue to consider only dimensional chain complexes. We have identi fied chains and cochains by the choice of a basis (the "basic" chains). Another method we have used to identify vectors and covectors is to introduce a metric (scalar product). We continue to represent cochains by matrices. We may introduce an arbitrary (positive definite) scalar product ( , ) in each of the chain spaces C p ' Given ( , ) and given a choice of basic chains in Cp we may
640
H A R M ON I C C H A I N S A N D K I R C H H O F F ' S C I R C U I T L A W S
then introduce, as usual, the "metric tensor" g ( ) j = ( Ei ' E j ) , yielding c') == ij . This inverse ci gi c'j = cTgc', and its inverse g (p) - I with entries g ( ) yiel ds a j j metric in the dual space of cochains, (ex , fJ ) = i g i j = g - I (The simplest c a se to keep in mind is when we choose basic chains and demand that they be declared orthonormal, i.e., when each matrix g is the identity. Thi s is what we effectively did in our previous section when considering the chain complex on the torus; ( E j , Ed was the identity matrix.) To the p-cochain with entries we may associate the chain with entries := Thus g ( p ) - I : C P -+ Cp "raises the index on a cochain" making it a chain, while g : Cp -+ C p "lowers the index on a chain" making it a cochain. We shall now deal mainly with cochains. If
pi p a b a T b.
(c,
(ai ) p (a j ) , aj g(p)jkak'(p) a chainn. c appears in a scalar product we shall assume that we have converted c to a cochai Let A : V -+ W be a linear map between vector spaces. The transpose A T is simply the pullback operator that operates on covectors in W * . A T : W*
-+
V*
If we were writing covectors as row matrices, A T would be the same matrix as as A but operating to the left on the rows, but since our covectors are columns we must now R R interchange the rows and columns of A , i.e., we write w R A i = A i W R = (A T ) / WR , and so
(Recall that in a matrix, the left-most index always designates the row.) Suppose now that V and W are inner product vector spaces, with metrics gv { g ( V ) i } and gw = { g ( W ) R s l respectively. Then the adjoint
=
j
A* : W
-+
V
of A is classically defined by (A ( v ) , w ) w = ( v , A* ( w » ) v . A * is constructed as fol lows. To compute A * ( w ) we take the covector g w ( w ) corresponding to w , pull this back to V* via the transpose A T g w ( w ) , and then take the vector in V corresponding to this covector, g v - I A T g w (w ) . Thus A* = gv - I A T gw . In components ( A * ) j R = g ( V ) j k ( A T h S g ( WhR = g ( V ) j k A S k g ( Wh R ' In summary A* = g v - I A T g w s A * j R = A R j : = g ( W ) Rs A k g ( V ) kj
(B.7)
Note that in this formulation A * would reduce simply to the transpose of A if bases in V and W were chosen to be orthonormal . The coboundary operator and matrix have been defined in (B.3), 8 P = ap+ 1 T . The adjoint 8* satisfies ( 8 (ex ) , fJ ) = ( ex , 8 * (fJ » ) . Then 8* 0 8* = 0
Consider 8 p : C p -+ C p + I . The metric in C p = C P * is the inverse g - I of the metric 1 + 1 ) . Since g (p ) in C p o Hence, from (B.7), 8 * = g ( p ) 8 T g ( p + 1 ) - = g ( p ) ap+1
(p)g(p
LAPLACIANS AND HARMONIC COCHAINS
(8 p ) * : C p + 1
-+
641
C P , we prefer to call this operator 8* p+ l .
(B .8) Thus in any bases 8 is a T , and in orthonormal bases 8 * = a.
B . d . Laplacians and Harmonic Cochains
We now have two operators on cochains and If a cochain a satisfies 8*a = 0 we shall, with abuse of language, call a a cycle. Similarly, if a = 8* {3, we say a is a boundary. We define the laplacian .6. : C P -+ C P by (B .9) or briefly .6. = 8 * 8 + 8 8 *
Note that .6. = (8 + 8*) 2 and .6. is self adjoint, .6.* = .6. . A cochain a i s called harmonic iff .6. a = O . Certainly a i s harmonic i f 8 * a = 0 = 8a . Also, .6.a = 0 implies 0 = ( (8 * 8 + 8 8 * ) a , a ) = (8a, 8 a ) + ( 8 * a , 8 *a ) , and since a metric is positive definite we conclude that 8*a = 0 = 8 a .
A :1C
cochain is harmonic ifand only ifit is a cycle and a cocycle.
(B . l O)
be the harmonic cochains. If y is orthogonal to all boundaries, 0 = ( y , 8 * a ) = ( 8 y , a ) , then y i s a cocycle. Likewise, if y is orthogonal to all coboundaries, then y is a cycle. Thus if y is orthogonal to the subspace spanned by the sum of the boundaries and the coboundaries, then y is harmonic. Also, any harmonic cochain is clearly orthogonal to the boundaries and coboundaries. Thus the orthogonal comple ment of the subspace 8Cp- l EB 8 * C p+ l is :1(P . A non-zero harmonic cochain is a boundary nor a coboundary ! For example, the cycle E l + E4 of section B .b cannot be a boundary. In our finite dimensional C P , we then have the orthogonal ("Hodge") decomposition Let
never
642
HARMONIC CHAINS AND KIRCHHOFF'S CIRCUIT LAWS
cocycles
cycles
r------� 1)CfJ- l
Figure B.2
Thus any cochain
fJ
is of the form (B . l l )
The three cochains on the right are unique (though a and y need not be). We can actually say more. The self-adjoint operator £:,. = 8 * 8 + 8 8 * has :J{' as kernel and clearly sends all of CP into the subspace :1(p1- = 8 C p - l EB 8 * C p + l . Thus £:,. : :1Cp1- � :l(p1- is I : I , and, since :J{p1- is finite dimensional, onto, and so £:,. : C P -+ :1Cp 1- is onto. Hence any element of :}Cp 1- is of the form £:"a for some a . E
andGivenisanyuniqfJue up tothterehe addiis antion ofasuchharmonithatc cochaifJn. fJ fJ fJ fJ -H (fJ) H (fJ) fJ fJ - H(fJ) fJ 8fJ H(fJ) Inis athunie cohomol ogy clcarsseprofesentcoacycltive.e ThefJ there q ue harmoni dimension of:XP is H a
:1(1-
a E C
£:"a =
(B . 1 2)
"Poisson's equation" £:"a = has a solution iff E ;](1- . Now let p-cochain and let be the orthogonal proj ection of into :1L Then :J{P.L and = £:"a = 8 8 *a + 8 * 8a
E
C P be any
is in
(B . 1 3)
refines (B . I I ). In particular, if is a cocycle, then, since the cycles are orthogonal to the cobound aries, we have the unique decomposition
=0
Thus,
::::}
fJ
= 88 *a +
(B. 1 4)
a
dim .
p
(B . I S)
.
K I R C H H O F F ' S C I R C U I T L A WS
643
8* z
There is a similar remark for cochains with = O. Since we may always introduce a euclidean metric in the space of chains C P ' we can say
1
(B . 6)
h 8h.
where a = 0 =
Inis athunie homol ogy classc roefpraecyclsenteatzivtehereh, i. e., q ue harmoni aandchain that is both a cycle and a cocycle, dim. Hp = dim. :KP = dim. H P
(B . I 7)
•
Three concluding remarks for this section. First, once we write down the matrices for a and = a T , the harmonic chains, the nullspace of �, can be exhibited simply by linear algebra, e.g., Gaussian elimination. Second, it is clear from the orthogonal decomposition (B . 1 6), that in the homology class of a cycle S For our toral example, and + are in the same homology class, since � and + is harmonic from (B .6). While it seems perhaps unlikely that + is "smaller" than 1 , recall that our basic chains are there declared orthonormal, and so = while + = Finally, we write down the explicit expression for the laplacian of a O-cochain
8
z, tEhe1 harmoni(E 1 c rE4)epr/e2sentative has the smallest norm, I l h l (E 1 E4)/2 2E 1 2 E 1 I1 2, I E 1 E411 1 ../2. 8 *¢o �¢ 8 *8¢
E4EIl z1 l . EE41 81 *8o¢, (B . I S)
B .e. Kirchhoff's Circuit Laws
Consider a very simple electric circuit problem. We have wire I -simplexes forming a I -dimensional chain complex with nodes (vertices) { Vj } and branches (edges) each edge endowed with an orientation. The vertices and edges are the basic 0- and I -chains. The circuit, at first, will be assumed purely resistive, i.e., each edge carries a resistance > 0, but there are no coils or batteries or capacitors. We assume that there is an
connect{eeAd} , eA
RA
i i i 2 , i4, i 7 •
external source of current (v j )
= j at each vertex vj
which may be positive (coming in), negative (leaving), or zero. In Figure B . 3 we have indicated the three non-zero external currents and The problem is to determine the current := in each edge after a steady state is achieved. Current is thus a real valued function of the oriented edges; it defines either a I -chain or cochain, denoted by I.
IA I (eA)
644
HARMONIC CHAINS AND KIRCHHOFF'S CIRCUIT LAWS
1'7 "'f ,
"
"
\
\
h
i4
.... /
/
I
I
I
1 '4
Fig ure B.3
{
In Figure B . 3 , Co has basis v " . . . , V7 } , C , has basis
",
"
\
\
\
\
{ e " ... e9}'
i(l') \
Figure B.4
Kirchhoff's current law KCL states that at any node v , the sum of all the currents flowing into v from the wire edges and the external source must equal that leaving. But (see Figure B .4) the edges coming into v form the coboundary of the vertex, and so 0 = I(8 v) + i(v) = aI(v) + i(v) . This suggests that the wire form a (since we are taking a boundary) and
currents I-chain i iJ
aI = -i
i
(KCL)
The external currents form a O-chain. We write I(e A ) = I A and (v j ) = Kirchhoff's voltage law involves the electric field in each wire. Let
�'(e) =
1 �, ' = 1 E · dx
•
e.
be the integral of the electric field over the basic I -chain This is the voltage drop along branch Since we are dealing with steady state, i.e., static fields, we know that the electric field I -form t.� ' is the differential of the electrostatic potential ¢ ; see (7.26). Hence = ¢ ( a e) = 8 ¢ ( ) This suggests that we should consider as a We have then Kirchhoff's voltage law
I-cochai&(ne.)
e.
e.
voltage
(KVL)
and the electrostatic potential at a vertex defines a O-cochain ¢ . Write �'(eA ) = &A and ¢ (vj ) = ¢j . ¢ is defined only up to an additive constant. Finally, Ohm's law says that the voltage drop across the resistor R is always RI. Since we are assuming at first that only resistances are present in each branch, we may A say l;� A = RA . (When batteries are present this will be amended; see (B .22). Since &.
I
645
KIRCHHOFF'S CIRCUIT LAWS
we i n t e r p ret t h e r e si s t a nces as det e rmi n i n g a in A AB g(1) A B RA oAB Weput the identity metric tensor in
is covariant and I is contravariant, C] , f, = g ( 1 ) I B . Thus the metric tensor in the I -chains is diagonal =
t�
=
metric (B . 1 9)
g ( 1 )I
Co ; thus the vertices { v j } are declared orthonormal and may be considered either as chains or as cochains. Kirchhoff's laws then yield, for the electric potential O-cochain ¢
From (B .9) we have (B .20) and from (B. 1 9) /1 ¢ = iH = -i (In circuit theory, 0 is called the incidence matrix and /1 the admittance.) If we can solve this Poisson equation for ¢ , then we will know �� in each Knowing this and the resistances, we get the current in each branch. Is there always a solution? From (B . 1 2) we know that a necessary and sufficient condition is that the O-cochain i of external currents be a boundary, i = 0* I (a 1 cochain f3 ) = where is the I -chain version of f3 . Let = L Then = = L Thus the sum of the - VA - ) , where ± are the vertices of L coefficients of all the vertices in the boundary of a I -chain vanishes. Conversely, in a chain complex that is connected (such as our circuit), meaning that any two vertices can be connected by a curve made up of edges, it is not hard to see that any collection of vertices with coefficients whose sum vanishes is indeed a boundary. We conclude
eA .
cA aeA oc,cA (VA + c
c eA ' cA eA . oc
VA
sts a solentutieorinntgo the circuitthequal e totasl exttThere ehe trnalotaexil current external current leaving, (B. 20) iff
L k i(vd = 0
(B .2 I )
which i s of course what i s expected. The solution ¢ i s unique up to an additive harmonic O-cochain. We claim that a harmonic O-cochain f has the same value on each vertex in our connected circuit. For if P and Q are any vertices, let c be a I -chain with boundary Q-P . Then f ( Q)-f ( P ) = = f = 0, since f is a cocycle. Hence, as to be expected, the potential ¢ is unique up to an additive constant. Just to illustrate the computations, consider a pair of resistances in parallel, Figure B . S . We know that we need to have = : = i o Put V I = [ 1 O ] , V2 = [O I ] T , = [ 1 O f , 2 = [0 I f , ¢ = [ ¢ I ¢2f and = [ o The matrix g ( 1 ) i s the 2 x 2 = [- l l f . diagonal matrix with entries R I and R 2 . We have = V2 - V I =
f(oc) o (c) i2 -ii l i -iO] T . ae l -
el
e
.
T ae2
646
HARMONIC CHAINS AND KIRCHHOFF'S CIRCUIT LAWS
Figure 8.5
Then
D. = -i becomes, from (B . I S ) , D. alg(1)-la/ =
-i
[- 11 ] [ I/oR I I IR2] [-- 11 11] ( l I R I I I R2) [ I - 11 ]
The laplacian matrix is -
Then
=
D.[[ 2 ]T
11
=
0
[ - i o io
=
+
V gives immediately
-
I
2 - 1
Since is the voltage in both branches, this gives the familiar result that the equivalent resistance for the two resistances in parallel is RI + Some words about First a simpli fication of notation. Since only the I -chains involve a non-standard metric (based on the resistances), we shall write rather than Let be the I -cochain, with the voltage of the battery in edge being positive if the direction from the negative to the positive terminal yields the given orientation of Consider a closed loop formed by a battery of voltage and a resistor across the poles of the battery. By Ohm's law the integral of .;� I over the resistor is RI But the integral of � I over the entire loop must vanish, and so the integral of .;; 1 over the battery part of the loop must be Thus when a battery is present in a branch we have, as expected, the voltage Kirchhoff's laws are then drop
circuits with batteries but no external( 1 1 currentIIsR. 2)- I . g g( 1 ). B BA eA, BA e ' A B R = B. = d -B. A eA ��A = RA[ - BA.
aI = O and & = gI - B = 8 and then D. al g-It� = aI - ag-IB -a[g-IB] D. = - a[g-IB]
(B .22)
=
=
(B .23 )
which always has a solution, since the boundaries are in J(1- . Note also that
[2R B gI-8.
which is Tellegen's theorem, saying that the total power loss in the resistors is equal to the power supplied by the batteries. Since is a Further, we note the following. Look at (B .22) , written as = I -cycle, = 0, its cochain version satisfies is thus the sum of a cycle and a coboundary and the two summands gl and are orthogonal. Thus, in Figure B .2,
B[
aI
gI
8*[gI] = O. B 8
I
647
K I R C H H O F F ' S C I R C U I T LA WS
tcyclhe cochai n versi o n o f i s t h e ort h ogonal pro j ect i o n n es, IA nA I A nAC n I A n AC 'nAC nCA .el. 12 n2 1 n21e l ; n l 2 12 ! sources KA I
c
B
of into the subspace of chain 1 {eA} e 2 e2 , eA ' constant current B
= Bc . Thus if we choose an orthonormal basis for the cycles, the "meshes," then given any battery cochain B we can easily project it orthogonally into the cycle space, and the resulting cochain is the current. (For the we may write = Bc .) This is a special case of Weyl's method of orthogonal projection. The orthogonal projection operator is self-adjoint and depends only on the metric, i.e., the resistances in the given branches. In terms of the basic l -chains we have = Be , where = Consider a circuit where there is only one battery present, of voltage V, in branch Then the current present in branch = Bl = is and look at the V . Remove this battery, put it in branch new current in branch I' l = V = This surprising result is a special case of
Green's Reciprocity.
Finally, we consider the modifications necessary when there are present in parallel with the resistors in each branch
Figure B.6
not
KA :lC(eA) A KA eA , f
We do consider the current source a new I -chain :1\.
fA
as forming a new branch;
= :'j(A : =
{KA}
forms rather
eARA 1 A -KaeAA. RAU A - KA),
If is the current in branch i.e., is the current entering at one node of and leaving at the other node, then the current through the resistor is now The voltage drop along the resistor is then, by Ohm's law, and thus = BA . Kirchhoff's laws become, since the total current entering a node is still 0,
(.�A RA U A - KA)
-
(�
and
=
Poisson's equation becomes
8¢
=
g(1
- :1l)
-
B
(B.24) (B .25)
Orthogonal projection onto cycles now says IA =
II
AC
(Bc + :1(c )
(B.26)
APPENDIX
C
Symmetries, Quarks, and Meson Masses
At the end of Section 20.3b we spoke very briefly about "colored" quarks and the resulting Yang-Mills field with gauge group SU(3). This was not, however, the first appearance of quarks. They appeared in the early 1 960s in the form of "flavored" quarks, independently in the work of Gell-Mann and Zweig. Their introduction changed the whole course of particle physics, and we could not pass up the opportunity to present one of the most striking applications to meson physics, the relations among pion, kaon, and eta masses. This application involves only global symmetries, rather than the Yang-Mills feature of the colored quarks. For expositions of particle physics for "the educated general reader" see, e.g., the little books [ ' t Hooft] and [Nam].
C.a. Flavored Quarks
The description to follow will be brief and very sketchy; the main goal is to describe the almost magical physical interpretations physicists gave to the matrices that appear. My guide for much of this material is the book [L-S ,K] , with minor changes being made to harmonize more with the mathematical machinery developed earlier in the present book. As to mass formulas, while there are more refined, technical treatments (see, e.g., [We, Chap. 1 9] ) applying (sometimes with adjustments required) to more mesons and to "baryons," the presentation given in Section C.f for the "0- meson octet" seems quite direct. Flavored quarks generalize the notion of the Heisenberg nucleon of Section 20.3a The symmetry group there, S U(2), is called isotopic spin, or briefly isospin. Isospin refers to the "internal" symmetry group S U(2) and is not to be confused with the usual quantum mechanical spin [Su, Section 4. 1 ] , which refers to the space symmetry group SO(3), but the terminology mimics that of ordinary spin. (Recall that SU(2) is the twofold cover of SO(3).) Thus since isospin for the nucleon has two states p and n, we say that these nucleons have isotopic spin = 1 /2. In general (number of states) = 21 + 1 . The diagonal normalized third Pauli matrix h = 0 /2)0"3 is, except for a factor
I
648
FLAVORED QUARKS
649
of R, an infinitesimal generator of SU(2) and is called the isotopic spin operator h p, being an eigenvector of h with eigenvalue 1 /2, is said to be the nucleon state of isotopic spin 1 /2, while the neutron is the state of isotopic spin - 1 /2. In the quark model the nucleon is no longer considered basic; it was proposed that nucleons and many other particles are composed of quarks. For our purposes, we need only consider particles at a given space-time point. (We shall not be considering kine matics nor quantum dynamics.) Associate with thi s point a complex three dimensional vector space Q, a copy of ((: 3 , with a given orthonormal basis and the usual hermitian metric ( z , w) = ZT w. A quark is represented by a unit vector
in Q. If m = (m 1 , m2, m 3 ) T is any vector in Q, then it defines a (complex) linear functional /L on Q by /L (w) = (m, w) = 'L m j w j . (We may use subscripts throughout since our bases are orthonormal.) Thus the covariant version of the vector m = (m 1 , m2 , m 3 l is the covector given by the row matrix /L = (m 1 , m 2 , m 3 ) . I f q = (u d S ) T i s a quark, then its covector q * = ( Ii d :5) is assumed t o describe the anti quark of q , written here as q * since its matrix is the hermitian adjoint of q . For formal "bookkeeping" purposes we will concentrate not on the individual quarks but on bases or frames of three quarks or antiquarks. Let u, d, and s be the basis vectors of the given Q. These three quarks are called the up, down, and strange flavored quarks associated with this basis. A second basis related to this one by an SU(3) change of basis will result in a new set of u, d, and s flavored quarks. These flavors are not to be confused with the colored quarks of Section 20.3b. A quark frame q of orthonormal vectors in Q,
q = [u, d, s1 is written as in geometry (p. 250) as a formal row matrix (formal because the entries are quarks rather than numbers.) Since the quarks u, d, and s are orthonormal, their three antiquarks u* , d* , and s* form an orthonormal basis for the dual space Q* and we can consider the formal dual frame of antiquarks,
It was assumed that the part of the Lagrangian dealing with the strong force is invariant under an SU(3) change of frame in Q. If, e.g., one observer believes the quark in question to be a down quark d, another could see it as an s. Thus, just as with the Heisenberg nucleon, u, d, and s are to be considered as three states of the same particle, the flavored quark. Invariance of the Lagrangian under the eight-dimensional group S U(3) led to Gell-Mann's denomination of this theory as the "eight-fold way," using a phrase from Buddhist thought.
650
SYMMETRIES, QUARKS, AND MESON MASSES
To view a nucleon a s composed o f quarks, quarks are assumed to have fraction al electric charges
Q (u) = 2/3 ,
Q (d) = Q (s)
=
- 1 /3
(C. I )
(Charge thus violates SU(3) symmetry, but recall that SU(3) symmetry is assumed only for the strong force, not the electromagnetic.) It turns out, e.g., that the proton is made up of three quarks, written p = duu, whose total charge is - 1 /3 + 2/3 + 2/3 = 1 . The neutron = ddu has charge O. The electric charge of an antiquark is always the negative of that of the quark. The antiproton = d*u*u* has charge - I .
p
n
p*
C.h. Interactions of Quarks and Antiquarks
A composite particle formed from a quark and it its antiquark q * is described by physicists by considering the tensor product * @ in Q* 0 Q. Recall that if e is a basis for a vector space Q and if () is the dual basis for Q* , then for a vector v = ej v j and covector a = we have a @ v = @ ej v j = j @ ej ) v and Q* @ Q thus has basis elements @ ej . Each basis element @ ej defines a linear transformation sending Q into itself, (v)e j = v k ej , 0 ej ) (v) = The formal matrix @ e with entries 0 e)k j = 0 ej forms a frame for Q* @ Q.
qq q k k a cr akcr k ak (crk cr' (crk cr' crk butcrk we shall largely ignore this aspect. cr (cr * We shal l he deal i n g ent i r e l y wi t h t h e formal aspect s o f al l t h ese mat r i c es. q is meractelys abyformal col u mn mat r i x , q i s row mat r i x , q q i s a mat r i x , and g(q* ) q and g(q) qg-I . [ ] q* q [ ] we have omit ed the tensor product sign that inofthtehepresent columncase,matrtihxeqte* nsorand product the row matmatrriixx q.is the same as the usual matrix product q What part i c l e or particles do the nine entries of the frame q q represent? g * same q ' linear * (tragqnsf) ormat(qgion. g q q) g *@ 3x3 SU(3) We are interested in antiqu ark-quark interactions forming composite particles. The appropriate frame is =
g
a
*,
=
@
=
u* d * @ [u d s 1 s*
=
u*u d*u s*u
u*d d*d s*d
u*S d*s s*s
In the 3 x 3 matrix on the right in each entry; e.g., u*u is really u* @ u: We are not interested in the fact that, e.g., the entries in the frame are themselves matrices Note also
This would not be the case for the product in the reverse order in which case the tensor product frame matrix would again be 3 x 3 while the matrix product would be a 1 x 1 matrix. The three entries u, d, and s of the row matrix are identified as the three states of the quark, while the entries in q * are the states of the antiquark. *@ Any quark q can be sent into any other quark q ' by some E S U (3) . This is why we consider the different flavors up, down, and strange as being different states of the @ q' = particle. The group G = S U (3) acts on the tensor product frame by I I @ - ) = ( *0 - i.e., by the adjoint action Ad( G) as it does on a ,
651
I N T E R A C T I O N S OF Q U A RKS A N D A N T I Q U A R KS
If any two antiquark-quark frame matrices A and B were necessarily related by a g E S U (3) , B = g Ag - l , then we could conclude that the nine entries in q * ® q are simply the nine states of a single particle. But this is not the case! Clearly the scalar matrices C = AI, Cij = A 8ij , form a one-dimensional complex vector subspace of the nine-dimensional ([: 9 (= space of complex 3 x 3 matrices) that is left fixed under the G action, gCg- 1 = C. We conclude that the states of at least two particles appear in the frame q * ® q. Since Ad (G ) acts by isometries on ([: 9 , the orthogonal complement of the scalar matrices must also be invariant, i.e., sent into itself by Ad(G). If D is orthogonal to I , then 0 = L ij oij Dij = tr I D, and so the orthogonal complement of the scalar matrices is the complex eight-dimensional subspace consisting of 3 x 3 matrices. (Clearly tr A = 0 iff tr gAg- 1 = 0.) We say that the adjoint action or representation of S U (3) on the space of 3 x 3 complex matrices is reducible, breaking up into its action on trace-free matrices and its trivial (i .e., identity) action on scalar matrices. We should remark that if we had been looking, e.g., at antiquark-antiquark in teractions, the frame 0" ® 0" would again be a 3 x 3 matrix with entry O" i ® 0") i i and would transform under G E �-S (3) to Gri O" ® G.,j O") = Gri O" ® O") GJ,, ; i.e., 0" ® 0" --+ GO" ® O"G T = GO" ® O"G , which (because of the complex conjugation). S ince A --+ G A G T preserves and this is the natural decomposition to use in this case. We now decompose every 3 x 3 matrix A into its trace-free and scalar parts,
trace-free
ij does notsymmet preservery tracesantisymmetry,
A = [ A - ( 1 /3) trA I] + ( 1 /3) trA I . I n particular, for the matrix q * ( 1 /3) tr q *
® q w e have the scalar part
®qI
= ( 1 /3) (u * u
+ d*d + s * s) I
(C.2)
and then the trace-free part becomes X : = q*
® q - ( 1 /3 ) tr (q * ® q) / (C.3)
1 - (2u * u - d*d - s* s) 3
u* d
u*S
1 - (-u * u + 2 d * d - s* s) 3
d*u s*u
s*d
d*s I - (-u * u - d *d + 2s * s) 3
Since the scalar matrix (C.2) never mixes with the matrix X under S U (3) we can use it to define a new particle, the eta prime, '11
'
:= ( l / J3) (u * u
+ d * d + s * s)
(C.4 )
Why does the factor 1 / .)3 appear? The quark flavors u, d, and S are unit vectors in Q, and likewise for the antiquarks in Q* . Thus u*u, etc. are unit vectors in Q * ® Q, and
the three vectors in (C.4) are orthonormal. The factor 1 / .)3 makes the r/ a unit vector. Since quarks and antiquarks have opposite charges, the 1] ' is a neutral particle. ['tHooft,
652
SYMMETRIES, QUARKS, AND MESON MASSES
p. 46] interprets the sum in (CA) a s implying that the r / is u * u d*d s * s". The nine entries of the matrix X of (C.3) can represent at most eight particles since the trace is O. To understand the action of = S U (3) on X we notice the followin g. G is acting by the adjoint action on the space of traceless matrices. Now S U (3) acts by the adjoint action on its Lie algebra
"continuously changing/rom
to to
G
B real a B. ib)B ibB all Thus we may consider our particle matrix as being in complexification C.c. The Lie Algebra of SU(3)
Physicists prefer hermitian to skew hermitian matrices, since observables in quantum mechanics are represented by hermitian operators. Note also that our matrix X is for mally hermitian. Gell-Mann chose for a basis of � := .J=T �u(3), i.e., the traceless hermitian matrices
AJ
=
A5 =
A,
�
[! [� [�
1
0 0 0 0 0 0 -1
0
�l �;l �l
A2
�
A6
![ -i �l [� !l [� � l
A4
0 0 0 0
�
=
A7 =
0
1 A8 = -
v'3
[� �l [� �;l 0 0 0 0 0
2
0
These matrices are orthonormal with the scalar product ( A , B ) : = ( 1 /2)tr A * = ( 1 /2) tr A B in 3" Note that A k , k = 1 , 2, 3 , are just the Pauli matrices with zeros added in the third rows and columns, and when exponentiated these A k } generate the subgroup S U (2) C S U (3) that leaves the third axis of ((: 3 fixed. Let us expand X = L J :'Oj :'0 8 X j A j , with all X j real. The only A with entry in the (3,3) spot is A 8 , and thus - ( 1 /3 ) (u * u + d*d 2s * s) = ( X 8 A 8 hJ = X 8 ( - 2 / v'3) and so X 8 = ( 1 /2 v'3) (u * u + d*d - 2s * s) = 1]/ � , where the particle 1] is defined by the unit vector
{i
B
-
1] := ( 1 / J6) (u * u + d*d
-
2s * s)
(C.S)
653
PIONS, KAONS, AND ETAS
Then TJ v'6
X8 A8 =
0
0
0
TJ
0
0
-2 TJ
v'6
0
v'6
Then from (C.3) we get for X
1 2
TJ - (u * u - d * d) + -
u*d
d*u
- � (u *u - d * d ) + -.!L
S*u
v'6
2
s*d
u*S v'6
d*s 2 TJ
v'6 Finally, we define three sets of particles (with explanation to follow) :
{ 1r0 = ( l / .J2) (u*u - d*d ) 1r- = u*d 1r + = d*u} K O = d*s} { K- = u*s KO = s*d} {K+ = s*u
(C.6)
and then
(C.7)
C.d. Pions, Kaons, and Etas
The seven particles listed in (C.6) and the eta in (C.5) have physical attributes that led to their identification in the particle world. First there is electric charge. For example 1r- = u*d has, from the quark charges (C. l ) the charge -2 / 3 - 1 / 3 = - 1 . This is the reason for the minus sign attached to the 1r symbol. Neutral charge is denoted by the exponent 0, as for example in 1r 0 . This explains the exponents in (C.6). Note that, e.g., 1r- is the antiparticle of 1r + while 1r0 is its own antiparticle. Physicists usually O denote antiparticles by a complex conjugation overbar. K is the antiparticle of KO and is distinct from KO , as we shall soon see. These eight particles are among those called mesons, because of their masses being intermediate between those of electrons and protons. The diagonal matrices diag{ e i e , ei¢, e -i(tI+r/l) ) form a two-dimensional, maximal connected subgroup of the eight-dimensional S U (3), i.e., a maximal torus T 2 . (The maximal torus of U (n) was discussed in Theorem ( 1 5 .4).) Note that the
commutative,
654
SYMMETRIES, QUARKS, A N D MESON MASSES
two generators A 3 and A 8 generate, b y exponentiation, two I -parameter subgroups of this torus. Thus A 3 and A 8 form an orthonormal, basis for the Lie algebra of T 2, the tan gent space �of T 2 at the identity. (The Lie algebra of the maximal torus of any Lie group G is called the Cartan subaJgebra Pv of 3") We now change slightly the normalization of four of the Gell-Mann matrices h :=
and
( l / 2) Ab
k = 1 , 2, 3
Y : = ( l / J3) A 8 = diag { I / 3 1 / 3 ,
(C. 8) ,
-
2/3 }
The I s generate the S U (2) subgroup of S U(3), call it SU ( 2 )
[S �( �]
x
1,
2)
called the isospin subgroup, and Y is the generator of the I -parameter subgroup of SU(3) called hypercharge. S ince the I s and Y are hermitian they represent "observables" ; since further h and Y commute they are "compatible" [Su, p. 57] , and so in a sense they can both be measured simultaneously. The flavored quarks are eigenvectors of these operators: 13 (d) = (- I /2)d
h ( u ) = h ( l 0 O) T = ( I / 2)u Likewise Y (u) = 0 / 3 ) u
Y (d) = 0 /3)d
Y es)
=
( - 2/ 3) s
Furthermore, if q = (u S ) T is a quark, then an infinitesimal generator A of SU(3), say A = 13 or A = Y , is basically a differentiation operator, i .e.,
d
dt I dt I dt I dt I
i A (q ) := � e iA I (q)
� e iA l q
=
1=0
or briefly
A (q ) = � (e I A q)
1=0
while if q *
=
=
i Aq
1=0
= Aq
(Li d s) is an antiquark
A (q * )
=
� (q * e - I A )
1=0
= -q * A
Thus h ( u * ) = - (1 0 0) h = ( 1 /2)u * . In general, if the quark q is an eigenvector of a Gell-Mann generator A then its antiquark q * is an eigenvector with oppositely signed eigenvalue. Finally, since each generator is a differentiation -
A (q
Q9
q')
=
A (q)
Q9
q' + q
Q9
A (q ' )
655
PIONS, KAONS, AND ETAS
Since u, d, s, u* , d*, and s* are eigenvectors of 1 3 and Y, any composite particle built up from them will also be an eigenvector whose eigenvalues are the sums of the constituents. For example, Y (KO ) = Y (s*d) = (2/3 + l /3 ) KO = KO while Y ( K ) = - K o . This shows indeed that K O KO Isospin and hypercharge play a very important role in describing particles. The eigenvalues of h and Y (briefly h and Y) are two numbers that one assigns to strongly interacting particles with the experimentally observed property that if several particles collide and become other particles, then the sum of the isotopic spins before collision is the same as after, and likewise for the hypercharge. These "conservation laws," together with Noether's conservation principle (20.9), suggest that both the isospin and the hypercharge groups might be symmetry groups of the strong force Lagrangian. This is the origin of the hope that SU (3) , which contains both as subgroups, might even be a large symmetry group, or at least an approximate one. In Figure C. l , we exhibit graphically h and Y for each of the representations of SU (3) that we have considered. The result will be called the weight diagram of the
o
and are distinct particles.
y
2/3
u
d
y
S
1T+
K Figure
C.l
y
S
656
SYMMETRIES, QUARKS, A N D M ESON MASSES
representation. Since h and Y are a maximal set o f (two) commuting operators, we shaH have a two-dimensional graph for each representation. In Figure e. l , the representation 3 is the standard representation of SU (3) on ([: 3 , i .e., on vectors q = S ) T . (Phy sicists label the representations by their dimension, with or without an overbar.) We have drawn Cartesian axes labeled 13 and Y and have placed the particle u at the poi nt with coordinates given by its eigenvalues h (u) = 1 /2 and Y (u) = 1 /3, etc. The next representation is the representation labeled by physicists 3; it is the repre sentation on the dual space ([: 3 * , i.e., on anti quarks q * = (Ii d s) . The eigenvalues here are the negatives of those in 3 and so the weight diagram is the reflection of that for 3 through the origin . We have also used the physicists' labels IT instead of u * , etc. The final diagram is that for 3 ® 3. There are three particles 7r0 , "I, and "I ' at the origin, requiring a point surrounded by two circles. Note that this diagram is easily constructed graphically from the two previous ones because of the additivity of the eigenvalues. To construct it we take the whole diagram of 3, translate it so that its origin is at a particle of 3 (say IT), erase that particle, and mark in the positions of the three particles of the translated 3; then we repeat this operation at the two remaining particles of 3. We have seen before that this representation is reducible, the particle "I ' being fixed under all of SU (3) . If we remove this particle (the one-dimensional space 1 of scalar matrices) we get the weight diagram of the adjoint representation, denoted by 8. It differs from that of 3 ® 3 only by having a point and one circle at the origin. Physicists say
(u d
(e.9) There is (at least) one serious problem remaining. Our eight particles - the three pions, the four kaons, and the single eta - had been matched up by the physicists with the eight observed mesons with those names. While the observed particles in each category (e.g. , the three pions ) have roughly the same mass, the masses of pions, kaons, and the eta differ widely. Since masses are coefficients that appear in the Lagrangian and the Lagrangian is assumed invariant under SU (3) , the assumption of S U (3) i nvariance will have to be modified. C.e. A Reduced Symmetry Group
The mass of a pion is observed to be 1 40 MeV, the four kaons are at 49S MeV, and the eta has a mass of SSO MeV. (In comparison, the electron mass is about 1 12 MeV.) This suggests that the strange quark S might be considerably heavier than the up and the down quarks. On the other hand the equality of the three pion masses sug gests that u and d have about the same mass. Individual quarks have never been seen; in fact there are reasons to believe that they will never be seen (quark "confinement"). It was then suggested that S U (3) is too large to be the symmetry group for the strong interactions. Experimentally, however, isospin and hypercharge a re conserved in strong interactions. This suggests that the isospin subgroup S U (2) x 1 and the I -parameter hypercharge subgroup U ( 1 ) = diag( ei l1 , e ili , e -2i ll ) of SU (3) generate a more realistic symmetry group. Since As commutes with Ab for k = I , 2, 3 , it is clear that the three
A REDUCED SYMMETRY GROUP
657
A k s together with A8 form a Lie subalgebra of �.u{3) and so, from ( 1 5 .34), generate a four-dimensional subgroup, call it S U (2) * U ( 1 ) , of S U (3 ) . We shall identify this group, S U (2) * U ( l ) consists of all products from the two sub groups, but since S U (2) x 1 and U ( 1 ) commute we need only consider the product of , e 2 i e ) E U ( 1 ) . Let a be a 2 x 2 matrix in pairs g E S U (2) x 1 and h = diag( i , SU (2) . It is clear that to each of the products ( i a ) x we may associate the U (2) matrix a, and in fact this correspondence S U (2) * U ( 1 ) -+ U (2) is a homomophism onto all of U (2) . The kernel consists of those a and e such that a = the 2 x 2 identity h Since det a = 1 we have the two-element kernel with a = ± h /z ; i.e., a = and s o the group S U ( 2 ) * U ( 1 ) can b e considered a s a two-sheeted covering group of U (2) . Let now G : = S U (2) * U ( 1 ) be assumed to be the symmetry group for the strong interactions . It has generators Ab k = 1 , 2 , 3 (isospin), and A8 (hypercharge) and all operate again on the quarks ([:3 , antiquarks ([: 3 * , and mesons ([: 3 * ® ([: 3 . Our basic meson frame is again X of (C.7). G can mix u and d but neither of these mixes with s. Thus we may consider u and d as two states of the same particle, but s is assumed to be a different quark, with only one state. A typical element of S U (2) * U ( 1 ) is of the form
ification wil play no further role in our discussion since only the generatbutors twihelidbeentneeded. e ll eill -e ll e-2ie ei 8 eill ie- ll
g
(C. l O) with I d + I w l 2 = 1 and the 2 x 2 submatrix in SU (2) . Consider the adjoint action of this matrix on X. Since this operation is linear in X we may single out the particles in which we are interested. For the antikaons K - and K ° we may take for X the matrix
[0 00 ] 00 o o
KKO
and we see easily that the adjoint action by (C. l O) will produce mixtures of the K- and the K o . Similar results can be obtained for the K+ and the KO . Since the (K- , K 0 ) do not mix with the (K+ , KO ) , we see that (K+ , KO ) are to be considered as two states of a single particle and (K- , K O ) are the two states of the antiparticle. (They are distinct particles, as we see from the weight diagram (Figure C . l ) that their hypercharges are opposites.) Similarly, all three pions get mixed; they are three states of a single particle. Finally, the eta is completely unaffected by the adjoint action. We say that (K+ , KO ) is O a doublet, its antiparticle (K- , K ) is a doublet, the pion (7r - , 7r0 , 7r + ) is a triplet, and 11 is a singlet.
658
SYMMETRIES, QUARKS, AND MESON MASSES
C.f. Meson Masses
A fermion is a particle (e.g. , an electron, proton, . . . ) whose wave function changes sign when an observer's coordinate system is rotated through a complete rotation (see p. 5 1 7) , whereas a boson (e.g., a meson) has a wave function that returns to its original value under such a rotation. Particles composed of an odd number of fermions are again fermions but an even number will yield a boson. A neutron, made of three quarks, is a fermion. This leads us to think of quarks as fermions. A kaon, made of two quarks, is a boson. Electrons and protons satisfy the Dirac equation, which can be "derived" from a Lagrangian (20. 1 8). The coefficient of the squared wave function 1 is the mass of the fermion in question. Bosons are believed to satisfy something similar to the Klein-Gordon equation ( 1 9.24). To get this from a Lagrangian the coefficient of 1 \11 12 must b e the square o f the mass, m 2 . (Actually there is also a factor o f 1 /2, but this will play no role in our discussion and so will be omitted.) We shall just accept the "rule" that
\11 2 m,
ient ofthe squared term 1 \11 2 in the Lagrangian involves mfor afermion and tmhe2 forcoeffia cboson. f mess component
The classification of the particles that we have given followed from looking at ra of quarks, antiquarks, and mesons, i.e., q, q * , and X. A Lagrangian involves (wave functions) rather than the basis elements (frames). For this reason we shall revert now to the component description of the meson matrix X, which formally is simply the transpose,
nO + Ji .j6 n K-
-
X=
17
n+
_n o
Ji
-
17
+ -
KO
.j6
K+ KO
-2 T}
.j6
where the entries now are components, rather than basis elements. For example,
u
(C. I I )
d.
K+
=
We are interested in the masses of the mesons. We shall postulate a mass part Lm of the total Lagrangian. In the original version, when SU (3) was assumed, we could use a quadratic Yukawa-Kemmer type Lagrangian involving our meson matrix X, namely L = tr X X * , but as we shall soon see, this would result in all the mesons having the same mass. For the symmetry group G = S U (2) * U ( 1 ) generated by isospin and hypercharge, we shall alter this by inserting an as yet to be determined 3 x 3 matrix W,
Lm
= tr X GJll X *
(C. 1 2)
To ensure that the mass coefficients are real we shall assume that ��ll is hermitian, for then X GJll X * will be hermitian and will have a real trace. Under a change of quark frame q used in Q = C 3 = EB C 1 , GJll is sent to where E G = S U (2) * U ( l ) . Since there is no preferred frame, we insist that 911 be unchanged under such a frame must commute with the G action on ([: 3 . It is then not change, and so ��11 : ([: 3 -+
([:2
([:3
g��l g -l , g
659
M E S O N M A S S ES
hard to see that ��ll
must
be of the form S'lll
=
diag (a ,
a,
b)
(C. 13)
where a and b are real numbers. I n fact, w e can apply elementary representation theory, in particular Schur's Corollary, as will be developed in Section D.c of Appendix D. An argument that is similar (but simpler) than that given there for the matrix C in that section, applied to C 3 = ([: 2 EEl C I rather than V = 5 EEl 1, will show that ��ll must be of the form (C. 1 3) . The key point is that the action of on ([: 3 leaves both ([: 2 and C I invariant and this action is not further reducible. ( I am indebted to Jeff Rabin for pointing out the uniqueness of this ��n.) We then compute, using the fact that formally X = X*,
G
Lm
=
�Jll
( I Jr 1 2 + Jr-Jr + + Jr + Jr - ) + ( l /3) ( + (K- K+)b + (K o KO)b + (K+ K-)a + (K o K O )
tr X
X
=
a
+
aa
°
Now the pion terms can be written
2b) 1 77 1 2
and the kaon terms as
b)[(K- K+) + (K o KO)] = O /2) (a + b) [I K+ 1 2 + I K- 12 + I K o l 2 + I K o I2]
(a
+
We have chosen this arrangement since all the kaons must have the same mass since K± are antiparticles, and so have the same mass, and since (K+ , KO) are the two states of a single particle (see the last paragraph of Section c.e.), and so have the same mass. Similar arguments follow for the pions. Then, since we are dealing with bosons,
mrrm'l mK
From these we see that
:=
mass of any pion
: = mass of eta
=
v
=
.J(i
+ 2b) /3]
[(a [(a
: = mass of any kaon
=
v
+
b)/2] (C. 1 4)
one of the famous Gell-MannlOkubo mass formulas. � 1 40 MeV and The masses of the pions and eta are � 550 MeV. Use these in (C. 1 4) . Then (C. 1 4) , i.e., = S U (2) * U0) (C. 1 2), yields the prediction � 48 1 MeV, which is less than 3% off from the observed 495 MeV.
observed mrr mry symmetryangroup G togetmherK with the simple choicethoefassumpt G-invariioanntofLagrangi
APPENDIX
D
Representations and Hyperelastic Bodies
D.a. Hyperelastic Bodies
In (A.2S) we have shown that the rate at which energy is stored in a body during a given deformation from reference body B (O) , assuming no heat loss, is given by
r S A B (dEA B /dt) VOL
(D. I )
JB (O )
where S is the second Piola-Kirchhoff stress tensor, E is the Lagrange defonnation tensor (2.69), and the integral is over the fixed reference body B (0) . The entire integrand is not necessarily the time derivative of a function . The stress tensor S is generally a complicated function of the deformation tensor E. In the linearized theory we assume generalized Hooke's coefficients C and a relation of the fonn (D.2) where, since both S and E are symmetric tensors, C is symmetric in A and B and also in J and K. At each point there are thus 36 constants C AB lK involved. Let us now assume the hypereiastic condition
C A B lK = ClK A B
(D.3)
Then at each point of B (O)
S A B (dEA B /dt) = C A B lK ElK (dEA B /dt) = d/dt ( I /2 C A B lK ElK E A B )
and then (D. 1 ) becomes
r S A B dEA B /dt VOL = d/dt r P B (O)
JB�
JB�
where
W
VOL (D.4)
A : = ( l /2P B (O) ) C A B lK El E A B = ( l /2P B (O) ) S B E A B K is the mass density of strain energy. A s mentioned at the end of Section A.e, a body
W
with such an energy function is called hyperelastic. 660
661
ISOTROPIC BODIES
Note that in this linearized case we have (D.S) where the partial derivative is taken while keeping the coordinates X of the reference body fixed. We remark that in the general (nonlinear) case of a hyperelastic body we may in fact use (D.S) to the stress tensor. That is, we can assume that there is some strain energy function W ( X , of the position X and of the the Lagrange deformation tensor and then use (D.S) to define the second Piola-Kirchhoff stress tensor S. This is compatible with the definition (A.27) that was used in Hamilton's principle, in this case of hyperelasticity.
define E) E Frommnowationonwiweth coeshalffilcirestentsriCctsatourisfyselinvesg to hyperel astpoiic bodi es in the linearized approxi at each n t . CAB
of independent body.
(D. 3)
Note that the number
JK is now reduced from 36 to 2 1 components at each point of the D.h. Isotropic Bodies
In the following we shall be concerned only with ]R 3 with an orthonormal basis. This allows us to forget distinctions of covariance and contravariance, though we shall frequently put indices in their "correct" place. In the linear approximation, S and are related as in (D.2). At each point we consider 6 the real vector space ]R of 3 x 3 matrices. (D.2) says that S E ]R6 is related 6 6 6 to E ]R by a transformation C : ]R ---+ ]R ,
E linear symmetric C(E) xC(E) E zE E'x . z E' E E' C(E') . E' g Eg-l ,
E
S=
(D.6)
Consider a given deformation tensor at a point. (For example, could result from a stretching along the axis and compressions along the y and axes at the origin. ) The result is a stress S = at the point. Now consider the same physical deformation but oriented along different axes; call it (In our example could be stretching along the axi s and compressions along the and axes, all with the same magnitudes as before. ) The new stress is S ' = If we call the change of axes matrix g E S O (3), then the matrices and are related b y = but w e must not expect S ' to be S - I ; the material of the body might react, say, to compressions along the and axes in entirely different ways. If we do have S ' = g Sg - l , in other words, if the 6 6 (adjoint) action of S O (3) on 3 x 3 symmetric matrices commutes with C : ]R ---+ ]R ,
y gy g
x
(D.7) and if this holds at each point of the body, we say that the body is elastically isotropic. We now have the following situation for an isotropic hyperelastic body. The real 6 x 6 6 6 matrix : ]R ---+ ]R has at most independent entries and the matrix commutes 6 with the adjoint action of S O (3) on ]R (thought of as the space of symmetric 3 x 3 matrices) . We shall sketch, in the remaining sections, how elementary representation theory shows that there are only two "Lame" constants required to express
C
21
C
C!
662
REPRESENTATIONS A N D H Y PERELASTIC B O D I E S
D.c. Application of Schur's Lemma
fJ., fini t e di m ensi o nal fJ., (g) fJ., (gh) fJ., (g)fJ.,we(h ).shallfrequently call the representatiogn
Consider a representation of a compact group G as a group of linear transformations of a vector space V into itself; see Section I S .2a. Thus, for E G, : V --+ V and = When we are considering only one re presentation fJ., of G on a vector space V , V rather than fJ., . We have in mind for our application, the following: Example: G S O (3 ) , V = ]R6 is the real vector space of symmetric 3 and /L (g ) acts on a matrix E by the adjoint action , /L (g ) ( E ) = g Eg - l . =
x
3 matrices,
Since G is compact, by averaging over the group (as in Section 20Ac), we may choose a scalar product in V so that acts on V by unitary or orthogonal matrices, depending on whether V is a complex or a real vector space. The representation is irreducible if there is no nontrivial vector subspace W that is invariant under all i.e., : W --+ W for all g E G . I f fJ., i s reducible, then there i s a nontrivial subspace W c V that i s invariant under G . I n this case the orthogonal complement of W i s also invariant since acts by isometries. Then by choosing an orthonormal basis for V such that the first dim( W) basis elements are in W and the remaining are in the orthogonal complement of W, we see that each is in block diagonal form. If fJ." when restricted to W, is reducible, we may break this reducible block into two smaller blocks. By continuing in this fashion we can reduce V to a sum of mutually orthogonal invariant subspaces, each of which forms an representation of G . I n our example V i s the space of symmetric 3 x 3 matrices The deformation tensor represents a covariant bilinear form and should transform as = but since = for in S O (3 ) , we may think of as a linear transformation = :ffi.3 --+ ffi. 3 . (This is nothing more than saying in an orthonormal basis). As a linear transformation, its trace tr will be invariant, and, just as we did for the meson matrix (C.3), we shall reduce the six-dimensional space of all symmetric 3 x 3 matrices into the sum of the trace-free symmetric matrices and its orthogonal complement of scalar matrices, which we could write in the same spirit as CC.9) as
fJ., (g) fJ.,fJ.,(g), fJ., (g)
fJ., (g) irreducible ElK g T g- l g
E [E =
g
E. fJ., (g)(E) gEgT , E l ElK E K
E
- ( \ /3) (tr
E)/]
+ ( l /3) (tr E) /
(D.S)
real Let fJ., ) and w) be t w o i r reduci b l e r e pr e sent a t i o ns o f actandionslonet and in thbee sensea lintearhat transformation that commutes with the A[fJ., (g )v] w(g) A [v]
where we are now indicating the dimensions. 1 is clearly irreducible, and we shall give a rather lengthy sketch showing that 5 is also. (V, A : V --+ W V W
Schur's Lemma: G
C W,
G
=
663
APPLICATION OF SCHUR'S LEMMA
1-1
Then ei t h er maps al l o f t o or i s and ont o . In t h i s l a t e r case we say that the representations JL and ware A
V
0 E W
A
equivalent.
The commutativity of A and the G actions shows immediately that the subspaces ker(A) c V and Im(A) c W are invariant under the G actions. Since V is irreducible, ker ( A ) is either V , in which case A ( V ) = 0, or ker A = 0, showing that A is 1 - 1 . In this last case, by irreducibility of W, we have Im(A) = W. 0
PROOF:
:
IJL(fJLgi),s iandrreduciCblhase andan eiaglenvect inear torarnsinformatthenionCCis a scalar commut e s wi t h each matrix, C AI. if
Schur's Corollary:
if
V,
V
--7
V
=
Note that i f V is complex, C will automatically have a n eigenvector. Let v be an eigenvector of with eigenvalue A. Then : V --7 V will also commute with the G action on V . But ) v = O. By Schur's Lemma, = O. 0
C
PROOF:
C -AI
(C AI C -AI
Return now to our elastic isotropic example. V is the space of real symmetric 3 x 3 matrices. : V --7 V is the linear map = in (C.6) relating stress to strain in the linear approximation. In terms of matrices, (0.7) says that C = commutes with the adjoint action of G = (3) on V . Since V is a real vector space, we must determine if has an eigenvector. But in the hyperelastic case, (D.3), i.e., = says that is a (symmetric) matrix operating on �6,
C S C(E)SAB C AB lKElK. SO C C AB lK CJK AB , C self-adjoint C irreduciAssume ble SO C(l) C(l) C(l) CC C(lC) C C a b. E)I S C(E) a [E E)I] JL A SAB 2JLEA B A(EJ )8AB
and so does have a real eigenvector. for the present that V = 5 EB I of (0.8) is a decomposition of V into subspaces, i.e., that the real, trace-free, symmetric 3 x 3 matrices form an irreducible representation of the adj oint action of (3) . We shall prove this in our following sections. Note that isotropy (0.7) shows that the subspace must be invariant under the G action. Since I is G invariant, the orthogonal projection TI of into 5 must also be G invariant. Since 5 is assumed irreducible, it must be that TI = 0 c 5, and so C (l) c 1 . Thus sends I into itself and, since is self-adjoint, : 5 --7 5. Then we may apply Schur's Corollary to the two cases, restricted to 5 and restricted to 1. In both cases C is a scalar operator. restricted to 5 is multiplication by a real number and when restricted to I is multiplication by a real From (D.8) we may write =
- ( l /3 ) (tr
=
+ ( l /3)b (tr
but this is classically written i n terms of the two Lame moduli =
C
and
as
+
which was essentially known already to Cauchy (see Truesdell [T, p. 306]) .
(0.9)
664
REPRESENTATIONS AND HYPERELASTIC BODIES
D.d. Frobenius-Schur Relations
Our only remaining task is to show that the trace-free, real, symmetric matrices 5 form an representation under the adjoint action of S O (3) . If our proof seems overly long it is because we are taking this opportunity to present very basic results about group representations. While our elasticity problem involves real representations, and real representations pose special problems (as in Schur's Corollary), we shall frequently use the notation of complex unitary representations (e.g., hermitian adjoint rather than transpose) but develop mainly those results that hold for real representations also, so that they can be applied to our problem. For more about the Frobenius-Schur relations, see, e.g., the small book of Wu-Yi Hsiang [Hs] (but beware that his Theorem 2 on p. 6 has been labeled Theorem 1 ) . The principal tool i s averaging over a compact group, as i n Section 20.4c. I f ( V , fJ.,) is a representation, then for each g E G , fJ., (g) is a matrix and its average, with respect to a bi-invariant volume form W G is again a matrix P : V ---+ V ,
irreducible
normalized so that the volume of is 1,
(D . l O )
meaning
P (v) for each vector v
E
=
fc fJ., (g) (v)wg
V . Clearly if fJ., (g)v = v for all g then P (v) = v. Also
fJ., (h ) P (v) =
fc fJ., (hg) (v)wg = fc fJ., (g)(v)wg = P (v)
(D. I I) (D.Il)
shows that P (v) i s fixed under all g , and so P : V ---+ V G , where V G i s the subspace of all vectors fixed under all fJ., (g) , the fixed set of the G action. Finally, from we see that
p 2 ( V ) = P ( P (v»
=
fc fJ., (h) P (v)wh fc P (v)w" =
=
P (v)
and so p2 = P; i.e., P is a of V onto the fixed subspace V G . Since this is a projection operator one sees immediately (by choosing a basis whose initial elements span V G ) that
projection
dim V G = tr P
=
fc tr fJ., (g)
(D. 1 2)
Wg
Let us look at some consequences of this formula. Let U and W be two vector spaces. Then U ® W * is the vector space of linear transformations of W into U ; ® w * ) (z) = w * (z)u , for all E W . Suppose that ( U , a ) and ( W, fJ) are representations of G on U and W respectively. Then the hermitian adj oint matrices fJ * (g) = fJ (g*) = fJ (g- l ) operate on W * by fJ * (g) (w * ) = W * fJ (g- I ) . Thus a ® fJ * is the representation sending the linear transformation A = ® w * to the linear transformation a (g)u ® w * fJ (g- l ) =
(u
z
u
665
FROB EN I U S - S C H U R RELATIONS
a ( g )A ,B ( g - I ) . A linear transformation A is fixed under this G action iff A commutes with the G action,
(U ®
W* )G
=
those A
:
W
--*
U such that a ( g ) A
=
A,B (g )
(D. l 3)
For representation /L the function X /J. : G --*
any W W,
tr (a ® ,B)
=
tr (a) tr (,B)
W,
since if {ej } and {fa } are bases for U and then {e j ® fa } is a basis for V ® W and the coefficient of e j ® fa in a ® ,B (ej ® fa) is a j j ,B0 a (no sum). Apply (D. 1 2) in the case V = U ® and use the fact that ,B ( g - I ) is the conj ugate transpose of ,B ( g ) . We get
W*, The di m ensi o n o f t h e s p ace o f t h at commut e wi t h the actions of is l ILemma n particthulisar,integral is and are irreducible and by Schur's ast one such map andOn tso,he otinhparter hand, icular, for andany War repreesentequiatvioalnent, there is atwelehave A: W
Theorem (D.14): G
--*
U
Xa (g ) Xp ( g ) wg
if C W, ,B) CU, a) O. if U
A
inequivalent,
l X/J. (g ) X/J. (g ) wg
C V, /L "I- 0), � I
is a representation and l then the representation is irreducible.
Theorem (D.1S): If ( V, /L)
X/J. ( g ) X/J. ( g ) wg = I
Suppose that ( V, /L ) is reducible. In Section D.c we showed that V can be Written as a direct sum of orthogonal, invariant, irreducible subspaces V = $ Va , and we can let /La be the restriction of /L to Va . A simple example to keep in mind is a representation /L of (which as a manifold is the circle with angular 4 coordinate e ) acting on V = ]R by two 2 x 2 diagonal blocks, where and are nonnegative integers : PROOF :
_ ( /L e ) -
SO(2) [m
c os e sin me 0 0
m
- sin e cos me 0 0
n Ln ]
0 0 cos e sin ne
-s cos e
SI m n
666
R E P R E S E N T A T I O N S A N D H Y P E R E L A S T I C B O D I ES
Call the 2 x 2 blocks ( VI , f.J.. I ) and ( V2 , f.J.. 2 ) ' The two representations f.J.. I and f.J.. 2 If = we would write V = 2 VI while if are equivalent if and only if =f. we would write V VI $ V2 . In the general case we can similarly write V Vj , where Vj and Vk are inequivalent if j =f. k. Then, from tr f.J.. = "L m tr f.J.. and (0. 1 4) we have
m
= m = n. m n =$ mj j j j
n
[ Xj.L(g)Xj.L(g)Wg = Lm j mk [ Xj(g)Xk (g)Wg = L m� [ Xj (g)Xj(g)Wg
Thus if f.J.. is reducible, i.e., "L We remark that a
complex
m
]
::: 2, we would have that the integral is ::: 2.
0
irreducible representation will have
[ X(g)X(g)Wg = 1
since by Schur's Corollary the matrices commuting with the G action will be scalar and so have complex dimension 1 . On the other hand, the usual action of S O (2) on 1R2 as in ( 1 5 .0) is clearly a real irreducible representation that has for integral of tr2
1
2](
4 cos 2
8d8/2Jr = 2
corresponding to the fact that the two-dimensional subspace of real 2 satisfying X22 X I I and X2 1 = X 1 2 all commute with S 0 (2) .
=
x
2 matrices
-
D.e. The Symmetric Traceless
3
x
3 Matrices Are Irreducible
(0. 1 5) implies that we need only show I tr Ad g l 2 wg r JSO(3 )
==g 5,g I
=
1
(0. 1 6)
where S O (3) acts on V the space of traceless real symmetric matrices, by the adjoint action, Ad(g) A A We have used before that S O (3) can be realized as the real projective space IR p 3 , pictured, e.g., as the solid ball of radius Jr centered at the origin of 1R3 with antipodal points on the boundary sphere identified; see Example (vii) of Section 1 .2b. The 1 parameter subgroups are the rays through the origin. This model i s unsuitable for the integral (0. 1 6) because in (0. 1 6) the metric is the same as the metric on IR p 3 , not 1R3 . 2 Since the unit sphere S 3 C ((: is the proper model for S U (2) (see Chapter 1 9 and also p . 5 84), and since S U (2) is the twofold cover of S O (3) , we shall use the "upper hemisphere" of S 3 as the model for IR p 3 . u For example, the point , 0 ) E S 3 C ((: 2 represents both the matrix E S U (2) and the matrix E S O (3), where -
u
i ( e ,8 g(,8 ) (,8 ) = [e-oif3 ei,8 ] 0
.
(,8 )
and
[COS 2,8 2,8 �Ol g(,8 ) = o2,8 sin 2,8 0
- sin cos
THE SYMMETRIC TRACELESS
3
x
3
MATRICES ARE IRREDUCIBLE
667
This was shown in the example following the proof of Theorem ( 1 9 . 1 2). We then have the following picture (see Figure 0. 1 ) on the unit sphere S 3 E ([: 2 with Riemannian metric d s 2 = det 2 + sin 2 et (de 2 + sin2 e d4J 2 ) , where et is the colatitude, and the "north pole" is the identity matrix for both SU (2) and S O (3), and the "small sphere" S 2 (et) at colatitude et has metric sin 2 a (de 2 + sin2 e dq / ) and area 47r sin2 a . We will explain this diagram more in the following.
_ _
S 2 ( a) (conjugates of g(�))
/-;
=
[COS
2� -sin 2� 0 cos 2 � 0 0 0 1
sin 2�
]
I I I I I I
= = -:.� - r-:;L-----�
_
g(�)
2
SO(3)
Figure 0.1
---
C
a=
�
2
=
�
The I -parameter subgroup u (f3 ) = diag(e- i fi , e i fi) C S U (2) , - 7r :::: f3 :::: 7r , is a maximal torus of S U (2) (see Theorem 1 5 .4), and the image of this circle under Ad : SU (2) � S O (3) (see Section 1 9. 1 b ) covers twice the maximal torus o f S 0 (3) given b y g (f3 ) , for - 7r /2 :::: f3 :::: 7r /2. The parameter f3 on this subgroup coincides with a for f3 :::: 0 and with a for f3 :::: O. (a is not a good coordinate at the identity.) For any point a of a Lie group G we can look at the conjugates of a , i.e., the set of all group elements of the form ga g - I as g ranges over the group. This set Ma is thus the orbit of the point a under the adjoint action of G on itself. The group elements that leave the point a fixed form the centralizer subgroup Co, of a , those g that commute with a . Thus, from ( 1 7. 1 0), the orbit points of Ma are in 1 - 1 correspondence with points of the quotient manifold Ma = G / Ca . Consider Figure D . l and the point a = g (f3 ) on the maximal torus. Since Adg : G � G sending any h to ghg- I is an isometry of the bi-invariant metric on G , and since Adg leaves the identity I fixed, Mg(fi) must lie on the sphere S 2 (a) at constant distance from I . It is not difficult to see (see Section E.a of Appendix E) that Mg(fi) in fact coincides with this 2-sphere. This is not surprising; the centralizer of g (f3 ) , f3 i- 0 or ± 7r /2, is exactly the maximal torus T I , and S O (3) / T 1 = S O (3)/SO (2) = S 2 . If f3 = 0, we have the identity I whose centralizer is all of S 0 (3) , and S O (3) / S O (3) is the single point I . If f3 = 7r/2, then the centralizer of diag(- I , - 1 , 1 ) contains not only the max imal torus T I (on which it lies) but clearly also the elements diag ( 1 , - 1 , - 1 ) and -
668
REPRESENTATIONS AND HYPERELASTIC BODIES
diag( - I , I , - 1 ) , which are rotations through 1 800 about the x and the axes respec tively. It is not hard to see, in fact, that all rotations through 1 800 about all axes in the xy plane are in this centralizer. This curve of rotations is the curve C' in Figure 1 7 .4. The conjugate set of diag( - 1 , - 1 , 1 ) is [ S 0 (3) / T U C'] . This is topologically lRP 2 , because SO (3) acts transitively on the space of lines through the origin of lR3 , and the subgroup leaving the axis invariant consists of all rotations about the axis, i.e., T , together with all rotations through 1 800 around all axes in the xy plane (i.e., C'). In our Figure D. l the conjugate set for (Jr /2) = diag( - 1 , - 1 , 1 ) is the equa torial 2-sphere with antipodal identifications, i.e., a projective plane ! (The conjugacy orbits M(J = G / Cr have very interesting topological properties in a general compact connected Lie group. For example, the Euler-Poincare characteristic of M(J is equal to the number of times M(J intersects the maximal torus, as we easily noticed with S2 and lRP 2 . See Theorem E.2 in Appendix E). We return now to our integral (D. 1 6). Recall that each Ad(O" ) is a 5 x 5 matrix. Look at a general point 0" in G = S O (3 ) . The character X has the property
y
z
z
g
is constant on conjugacy orbits. g(O) whose centralizers are larger than itse(f
That is, X Thus our function X Ad (O" ) , the trace of the 5 x 5 matri x Ad(O" ) , is constant on each of the 2-spheres S2 (a) of constant colatitude a In our volume integral, the two conjugacy sets at a = 0 and a = Jr /2 can be omitted. Note that these conjugacy sets to be omitted are precisely those passing through the only two points and g (Jr /2) of T T We can then evaluate our integral as follows, thanks to the fact that each remaining conjugacy sphere Mg(fi) meets T orthogonally: (D. I 7) We integrate only from 0 to Jr /2 (i.e., only half of the maximal torus) to avoid counting the spheres S2(f3) twice. The factor Jr - 2 is required since the Frobenius-Schur relations require that the volume of G must be normalized to unity, and the total volume of our SO (3) is
(' / 2 4Jr sin2 (f3)df3
io
=
Jr 2
We now need to know the character function X of Ad g (f3 ) along the maximal torus. A straightforward way is as follows. Write down a basis j , 1 .:s j .:s 5 , of the real trace-free symmetric 3 x 3 matrices, starting say with = diag ( I , - 1 , 0) . For o n the maximal torus, compute = and take L This calculation yields the result X Ad
E EJ g(f3)Ej g( -fJ) L Ej aij (f3), g(f3) = 4 cos 2 2f3 + 2 cos 2f3 - 1
Finally our integral (D. 1 7) becomes (with help, e.g., from Mathematica) 1 Jr
2
lJC/2 1 4 cos2 2f3 0
+ 2 cos 2f3 - 1 1 4Jr sin (f3)df3 = 1
2
2
aj . g(f3)
THE SYMMETRIC TRACELESS
3
x
3
MATRICES ARE IRREDUCIBLE
669
showing indeed that the representation 5 of 3 x 3 real symmetric trace-free matrices is irreducible. 0 One final remark should be noted. The character can be more easily computed by "general nonsense." Consider the following vector spaces of 3 x 3 real matrices: 3 ® 3 = all 3 x 3 matrices 3 0 3 = symmetric matrices 3 /\ 3 = skew-symmetric m atri ces 5 = trace-free symmetric matrices 1 = scalar matrices
-+
Then 3 181 3 = 3 0 3 $ 3 /\ 3 = (5 $ 1) $ (3 /\ 3) . But the Hodge star operator sends 2-forms to I -forms, * : 3 /\ 3 3. In an orthonormal basis of ffi? , the star op erator clearly commutes with the actions of S O (3), which shows that 3 /\ 3 and 3 are equivalent representations, 3 /\ 3 = 3. Taking traces of the representations, we get (tr 3) 2 = tr 3 181 3 = tr 5 + tr 1 + tr 3. Thus
XS = (X3) 2 _ X3 - X1 = [2 cos 2,B + l f - [2 cos 2,B + I ] - 1 = 4 cos 2 2,B + 2 cos 2,B - 1 which agrees with our previous calculation of X Ad g ( ,B ) .
APPENDIX
E
Orbits and Morse-Bott Theory in Compact Lie Groups
E.a. The Topology of Conjugacy Orbits
We now wish to study in more detail the topology of conjugacy orbits in a compact Lie group G with given maximal torus T . But first we present an example (more complicated than the S O (3) case of Figure D. 1 ) to keep in mind. Let G be the nine-dimensional unitary group U (3) . The subgroup of diagonal ma trices T = {diag[exp(i8d , exp(i82 ) , exp (i83 ) ] } is a three-dimensional maximal torus. Consider the diagonal matrix a = diag( 1 - 1 , I ) . The subgroup Ca that commutes with a, the centralizer of a, is U (2) x U ( 1 ) , which has dimension 4 + 1 = 5 . The conjugacy set of a, Ma = { u a u - I } is, from ( 1 7 . 1 0), in 1 : 1 correspondence with the complex projective plane C p 2 = U (3)I U (2) x U ( l ) , the analogue of the real pro jective plane discussed in Section 1 7 .2b. It has dimension 9 - 5 = 4. This orbit Ma consists of unitary matrices with eigenvalues - 1 , - 1 , and + 1 . Thus Ma meets T in the three points a, diag ( - 1 , I , - 1 ) and diag( l , - 1 , - 1 ) , the distinct permutations of the diagonal entries of a, and, as we shall see in Theorem E.2, the Euler characteristic X ( C p 2 ) is 3. The same argument would hold for any diagonal r = diag(e i 8 , e i 8 , e i
,
g- I ,
-
670
671
THE TOPOLOGY OF CONJUGACY ORBITS
1 : 1 , for if a - l = F ( C ) = F ( C ) = then ( - l E C, g E and so the coset is the same coset as
)a
= a(
l ) says that
hgC hah-l h ghC. h- g f : ]R ]R2 f. (¥r) � t t g(te e e e t e
g g g h-l g hC,
We now wish to show that this image Mrr i s an embedded submanifold. We show first that the differential F. maps no nonzero tangent vector to G / Ca at the single point aC into a zero tangent vector at the image point a ; i.e., that F is an immersion at a C . A n example to keep in mind about failure o f immersions is the map -+ given Y = which yields a curve with a C USp at the origin. This smooth by x = map is not an immersion because (�) = ( �� ) %x + vanishes at = O. This is the reason that a cusp can appear. S ince G j C is made up of curves -+ ) C , a general tangent vector at a C i s the velocity vector of a curve of the form l Y C, where Y i s i n the Lie algebra ;r of G . The image of this curve under F is F ( I Y ) = I Y a - I Y , whose velocity vector at = 0 is Ya - a Y, and this, by the definition of the differential, is F. (velocity of l Y C) . Suppose then that this Ya - a Y = o. Then Y = a Ya - l and so = ex p (a Ya l ) = a exp( y ) a - 1 (from the power series). Thus the curve exp(t Y) in G lies in Ca and so the curve l Y C is a single point curve aC and has zero velocity at = O. Thus F is an immersion at a C . This implies that F is an embedding of some G / C neighborhood of a C. But since each map Adg mapping G -+ G defined by -+ is a diffeomorphism sending Ma onto itself, it is not hard to see that F is locally an embedding near every point of G / C. Since G is compact, the situation pictured in the second curve in Figure 6.7 cannot arise. It can be shown that Ma is a global embedded submanifold of G . We now know, for a E that Ma is a submanifold of G of dimension dim G / Ca = dim G - dim Ca � dim G - dim T, since C Ca . Thus dim + dim Mrr � dim G . We shall accept the fact that every conjugacy orbit Mh must meet the maximal torus In the case of with maximal torus
(t) t 2 , (t) t 3 ,
te exp(t Y) t t h ghg-l
T,
T.
U(n ),
T
=
T
T
{diag (exp(i 8 1 ) ,
.
.
•
, exp (i 8 ) }
n
this i s just the statement that every unitary matrix can be diagonalized, i.e., for every E there is a g E such that is diagonal, i.e., in T . Thus each conjugacy orbit is of the form Ma , where a E Note also that our computation here has shown the following lemma.
u U(n)
U(n)
gug-1T. tangent space to at e is mapped andTheontoortthheogonal tangentcompl spaceement to toatthe under hogonaloflyin, tandersectisioevenn poidinmtsensiof onaland, and the Euler characteEachristic meetiss tTheortnumber T e.
L emma (E.1): 1: 1
Ma
Ya - a Y.
Theorem (E.2):
Ma X (Ma )
a
Y
Crr
-+
fr e t Y a e - t Y 1 0
=
Ma
T.
P R O O F : Let a E c Ca . Let Y be orthogonal to Ca at Then, in the bi invariant metric in G, Ya and a Y are orthogonal to Ca at a. We conclude from
672
ORBITS AND MORSE - BOTT THEORY IN COMPACT L I E GROUPS
Lemma E. I that Mer is orthogonal t o Cer a t (j . A schematic picture is given in Figure E. I .
e
y
Figure E.1
We now compute the Euler characteristic of Ma by means of the Poincare Hopf theorem ( 1 6. 1 2), using an argument that is a variation on ideas used by Weil and by Hopf and Samelson in the I 930s and I 940s . Let W be a tangent vector to T at the identity and consider the resulting I -parameter group of isometries on G, -+ = The velocity Killing field at any E G is w : = W W . O f course this field is tangent t o each o f the conjugacy orbits, in particular Ma ' Where are the zeros of this field w on G ? As computed previously, w = 0 at implies = and so g is in the centralizer of for all Now we may choose the tangent vector W to T s o that the I -parameter group lies dense on T (see Section 6.2a). For this W, E G is a zero if and only if is in the centralizer of the entire maximal torus T. It can be proved that T T C (T ) = T, see, e.g. , [Hs, p. 45] . For this W the zeros of the associated velocity field w make up the entire maximal torus T . In particular, the zeros of the Killing field w on Ma are the points where Ma meets T , these points being isolated since Ma meets T orthogonally. What then is the Kronecker index of the field w on Ma at such a meeting, say (j ? Since the I -parameter group is a group of isometries leaving (j fixed, the flow lines on Ma near (j must be tangent to a small geodesic codimension I -sphere S on Ma centered at (j . S ince w is a nonvanishing tangent vector field to this sphere S , S must have Euler characteristic 0, and so S must be odd dimensional and Ma is even dimensional. By 8.3( 1 1 ) the index of w at (j is + 1 and the sum of the indices at the zeros of w on Ma is exactly the number of intersection points of M" and
t g g( t ) et W gew• g-g g etWg get W , g ofa maximal torus is exactly itself, g (t)
T.
D
g
et W et Wt. the centrag lizer
APPLICATION OF BOTT' S EXTENSION OF MORSE THEORY
673
E.h. Application of Bott's Extension of Morse Theory
We conclude with some remarks concerning how the topology of the orbits Mer is related to that of the entire group G. For this we shall use Bott's refinement of the presentation of Morse theory that was given in Section 1 4.3c. For simplicity we restrict ourselves to the example U (3) with which we started our discussion, but similar remarks hold for all the "classical groups," U (n) , S O (n ) , Sp (n) , 2 (but not S U (n)), with some modifications ; see [Fr2] . The elements g of order 2 , g = I , are exactly four orbits, MI = I , M_ I = - I , and the two complex projective planes Ma and MfI , where a = diag ( 1 , 1 , - 1 ) and f3 = diag ( 1 , - 1 , - 1 ) . It is shown in [Fr2] that these points are also exactly the critical points of the function f (g) = Re tr( g ) , the real part of the trace of the unitary matrix g, and we can call these the "critical orbits." For Ma and M(3 , these are not isolated critical points but rather connected "nondegenerate critical manifolds" and one can apply Bott's extension of Morse theory (see, e.g, [Bo, Lecture 3]) to this situation. Briefly, we require that the hessian matrix for f be nondegenerate for directions orthogonal to the critical manifold. At a point m of Mer we can look at the part of the tangent space to G that is normal to M" and note the number of independent normal directions from m for which f is decreasing, i.e., the dimension of the subspace on which the hessian form is negative definite. These directions span a subspace of the normal space to Mer at m to be called the negative normal space. From nondegeneracy the dimension of these negative normal spaces will be constant along Mer and will be called the (Morse-Bott) index A( (T) of the critical manifold M" . The collection of all of these subspaces at all m E M" form the negative normal bundle to Mer . We ask that this bundle be orientable, meaning that the fibers can be oriented coherently as we range over the base space Mer . (If they are not orientable, we may proceed but we may only use £:2 coefficients when talking about homology groups.) Look at the point ex = diag ( 1 , 1 , - 1 ) at which f = 1 . Then the entire portion of the centralizer Ca given by U (2) x (- 1 ) c U (2) x U ( 1 ) , except for ex itself, lies in the region of U (3) where f < 1 = f (ex ) ; U (2) x ( - 1 ) is "hanging down" from the critical point ex. Thus there are dim U (2) = 4 independent directions at ex along which f decreases. Since f is invariant under a -+ gag- I , we see that this is true along all of Ma , and so A (a ) = 4. Similarly, the centralizer of f3 is U ( 1 ) x U (2) , the portion U ( 1 ) x diag( - 1 , - 1 ) hangs down from f3, and so the index in this case is A (f3) = dim U ( 1 ) = 1 . Of course I is the isolated maximum point and - l is the isolated minimum, and so A U ) = dim U (3) = 9 and A ( - I ) = O. Nondegeneracy can be proven for each of these critical manifolds. We shall need some classical results about the topology of the complex projec tive plane C p 2 , but we shall be very brief. Since C p 2 = U (3 ) / U (2) x U ( 1 ) , it is a compact 4-manifold. Recall from Problem 1 .2(3) that it is a complex manifold of com plex dimension 2. To all points in C p 2 with local homogeneous complex coordinates [zo , Z I , Z2 ] , where Z2 i- 0, we may assign the pair of genuine complex coordinates ( W I = ZO / Z2 , W2 = Z I / Z 2 ) ' For example, these will be local coordinates near the point [0, 0, I ] that represent the complex line along the Z2 "axis" (a copy of the ordinary Z2 plane). (Recall that [zo , z " Z2 ] i- [0, 0, 0] represents the complex line through the
674
ORBITS A N D MORSE - BOTT THEORY I N C O M P A C T L I E G ROUPS
origin of C 3 that passes through the point (Za , Z I , Z 2 ) ' ) We use the schematic picture in Figure E.2.
Figure E.2 The locus Z 2 = 0 consists of all points [Za , Z! , 0] with Za , Z ! not both 0, and thus is a complex projective line C p l (i.e., a 2-sphere; see Problem 1 .2(3)) with homogeneous coordinates [Za , z I J . We have a projection map h : C p 2 _ [0, 0, 1 ] --* C p l = S 2 defined by [Za , Z I , Z 2 ] --* [Za , Z ! , 0] . The locus I W I 1 2 + I W 2 1 2 = 1 represents a 3 -sphere S 3 in C p 2 centered at [0, 0, 1] and h , sending S 3 --* C p l = S 2 by ( w I , W 2 ) = [W I , W 2 , 1 ] --* [ W I , W2] , is simply the Hopf map of Section 22.4c. Note also that h : C p 2 - [0, 0, 1 ] --* C p l = S2 i s the endpoint of a deformation h t ( [za , Z I , Z 2 ] ) = [Za , Z I , ( 1 - t ) Z 2 ] that deforms C p 2 - [0, 0, 1 ] onto C p l . Thus any cycle on C p 2 - [0, 0, 1 ] is homotopic to one on the subset C p l . In particular, any singular j -cycle on C p 2 , for j < 4, can clearly be pushed slightly to miss [0, 0, 1 ] and can then be deformed into C p ! = S 2 . But S 2 is simply connected. Thus, since S 2 has nontrivial homology only in dimensions o and 2, we have the following:
Lemma ( E .3) : Any loop on Cp 2 is homotopic to a loop on S 2 and is thus de formable to a point; hence C p 2 is simply connected and therefore orientable. Hj (C p 2 , Z) = O for j = 1 and 3, and H2 (C P 2 ; Z) = H2 (S 2 , Z) = Z. Since C p 2 is a compact, orientable 4-manifold, H4 (C P 2 ) = Z. Since C p 2 is simply connected, the negative normal bundles to MOl and MfJ are orientable. Note also that by pushing S 3 by means of the deformations h I we may move S 3 to S; that lies at a small distance E: (in some Riemannian metric) from S 2 (see Figure E.3). B ut the points of C p 2 that are of distance S E: from C P I , for E: sufficiently small, form a normal 2-disc bundle N 4 over C p l in C p 2 , and the boundary a N forms a normal
APPLICATION OF BaTT'S EXTENSION OF MORSE THEORY
675
2-disc
,, circle, radius
£
,,
,,
"
,,
Figure E.3
Cp 2
,,
,,
,
[0 , 0 , I ] '\I
circle bundle (circles o f radius 8 ) . Thus S; = a N is this normal circle bundle to C p ' ' a N -+ C P is a realization of the Hopfmap S; -+ S 2 . Recall (see Section 1 4.3c) that the Poincare polynomial of U (3) is the polynomial in t with coefficients the Betti numbers bj of U (3 ) , and so PU (3 ) (t) = "L, bi t i . Bott's generalization of the Morse polynomial is constructed using the Poincare polynomials and indices of the critical manifolds (where we write Pa for the Poincare polynomial of Ma )
and the deformation h :
01IB (t) = 1 + tA (fJ } PfJ (t) + tA (a } Pa Ct) + t9 (Note that if each critical manifold reduces to an isolated critical point as in the original Morse case, then because bj (point) = 0 for j > 0, the coefficient of t A is again simply the number of critical points of index A . ) We have seen in Lemma E.3 that the Poincare polynomial o f Cp 2 is 1 + t 2 + t 4 , and so
Bott's generalization of the Morse inequalities are then 911B (t) ::::: PU (3 ) (t), but in [Fr2] it is shown that in U (n), the "symplectic" groups Sp (n), and S O (n), these are in fact equalities (except that one must use Z 2 Betti numbers in the case of S O (n), for n > 3 , since, e.g., the negative normal bundles to real proj ective spaces are not always orientable). Thus the critical orbits Ma yield exactly the Betti numbers of the group, but with each k cycle on Ma yielding a [k + A (a ) ] cycle on G .
"Final Exam"
If students find things that are worthwhile in this book, it is largely due to what I have learned from my own "teachers," among them Aldo Andreotti, Raoul Bott, S . S . Chern, Jim Eells, G.c. Evans, Harley Flanders, Heinz Hopf, Charles Loewner, Hans Samelson, Norman Steenrod and John Archibald Wheeler.
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Index
absolute temperature, 1 87 acceleration, 4-vector, 1 94 accessibility, 1 8 1 , 1 82 accumulation point, 1 06 action, 1 52, 274, 524 euclidean, 55 1 first variation of, 1 54 group, 454 Hamilton's principle of stationary action, 1 54 Jacobi 's principle of least action, 28 1 relativistic, 1 96 Ad, 486 bundle, 487, 489 connection, 487 adiabatic distribution and \eaf, 1 83 process, 1 80 adjoint, 392, 640 group, 486 representation, 486 admissible boundary form, 378 admittance matrix, 645 affine connection, 242 group of the line A( I ), 394 parameter, 272 Aharonov-Bohm effect, 447-8, 554 Aharonov-Susskind and spinors, 5 1 7 algebra homomorphism, 78 Ampere-Maxwell law, 1 2 1 , 1 63 angular momentum, 620 annihilator subspace, 1 67
anticommutator, 478 antiderivation, 89, 1 35 antisymmetric, 66 antiquark, 649 associated bundle, 482 connection, 483-7 Atiyah-Singer index theorem, 465 atlas, I S Bernoulli's theorem, 234 Berry phase, 468-72 equation, 472 Bertrand-Puiseux and Diguet, 288 Betti numbers, 1 57, 346 Bianchi identities, 300, 489 bi-invariant connection on a Lie group, 580 forms on a Lie group, 5 6 1 Riemannian metric and their geodesics, 563 binormal, 1 96 Bochner's theorems, 374, 530 Bonnet's theorem, 229 boson, 658 Bott's version of Morse theory, 673, 675 boundary (of a manifold) = edge, 1 06 boundary group, 344 homomorphism, 338, 60 1 operator, 335 boundary conditions essential or imposed, 527 natural, 527 683
684
bracket anticommutator, 478 commutator, 408 Lagrange, 80, 1 00 Lie, 1 26, 402; of �rvalued forms, 477 Poisson, 1 54 Brillouin and the stress form, 635 B rouwer degree, 2 \ 0- 1 3 , 360 fixed point theorem, 2 1 7 bump form, 1 07 bundle associated, 482 complex line, 433 cotangent, 52 determinant, 487 dual, 482 electromagnetic, 44 1 fiber, 4 1 5 frame, 453 gauge, 490 line, 433 local trivialization, 4 1 7 monopole, 444, 473 normal, 4 1 9 orientable, 6 1 1 principal, 454, 48 1 product, 4 1 8 projection, 4 1 5 pull back, 622 section, 50, 4 1 6, 466 space, 4 1 5 structure group, 433, 452 tangent, 48 transition functions, 24, 254, 4 1 4 trivial, 4 1 8 uni t tangent, 5 1 vector, 4 1 3- 1 7 volume, 488 canonical form, 394 canonical map, 1 49 Caratheodory's formulation of the second law of thermodynamics, 1 8 1 theorem, I 82 Cartan's bi-invariant forms, 562 exterior covariant differential, 250, 430 method for computing curvature, 257 structural equations, 249 theorem Jrz (G) = 0, 606 3-form on a Lie group, 566 H. Cartan 's formula, 1 35
INDEX
Cauchy equations of motion, 6 1 8, 620, 628 -Green tensor, 82 -Riemann equations, 1 5 8 , 1 59 stress form, 6 1 9; Lie derivative of, 634, 635 ; symmetry of, 62 1 center of a Lie algebra, 580 of a Lie group, 565 centralizer, 667 chain complex, 636 chain group, 337 integer, 336 simplicial, 343 singular, 333 character, 665 characteristic cohomology class, 6 1 6 charge form, 1 1 8 Chern's forms and classes, 587-9 1 ; as obstructions, 608- 1 6 integral, 6 1 2 proof of Gauss-Bonnet-Poincare, 462-5, 553-7 theorem, 6 1 5 Chern-Simons form, 586 Chern-Wei! theorem, 589 Chow's theorem, 1 7 8, 1 87 Christoffel symbols, 229 circulation, 1 44, 377 Clairaut's relation, 530 classical force, 1 95 momentum, 1 94 velocity, 1 93 Clifford algebra, 500 embedding, 262 numbers, 503 closed form, 1 56, 1 5 8 manifold, 1 20 set, I I closure, \ 06 coboundary, 638 cochain, 638 coclosed, 370 cocycle, 639 Codazzi equation, 229, 302, 3 1 1 - 1 3 , 320 codifferential d * , 364 codimension, 6 coefficient group, 337 field, 343
INDEX
cohomology H P , 356 integral class, 6 1 5 commutative diagram, 338 commutator bracket of matrices, 408 compact, 1 3 completable relative cycle, 387 complex analytic map, 1 5 8, 2 1 4 line bundle, 433; connections, 434 manifold, 2 1 composing rotations, 499 configuration space, 9, 50 conformally related metrics, 53 1 conjugate point, 327 conjugates, 667 connected space, 347 connection, 242 coefficients of, 243 , 429 curvature of, 244 electromagnetic, 440 fiat, 260 forms w, 249, 256 forms w* in the frame bOndle, 462, 480 induced, 309 Levi-Civita or Riemannian, 242, 245 on a Lie group, 580; fiat, 5 8 1 o n a vector bundle, 428-3 1 on the associated Ad bundle, 486 Simon, 472 spinor, 5 1 8-2 1 symmetric, 245 torsion of, 245 torsion-free, 245 constraint holonomic, 1 75 nonholonomic, 1 75 continuous, 1 2 continuum mechanics, 6 1 7-35; equilibrium equations, 630 contractible to a point, 1 6 1 contraction, 89 contravariant tensor, 59 vector, 23 coordinate change of, 29 compatible, 1 5 frame, 243 homogeneous, 1 7 inertial, 1 92 local, 3, 4, 1 3 map, 20 patch, 20
685
coset space G / H , 456 fundamental principle, 457 cotangent space, 40 coupling constant or charge, 539 covariance, 430 covariant components of a tangent vector, 43 constant, 267 derivative '\lx , 235, 24 1 ---4 , 430; second, 30 1 ; of a tensor, 298-9 differential '\l, exterior, 248 tensor, 58 vector = covector, 4 1 covector, 4 1 transformation law, 42 covering space, 569-76 associated to a subgroup of IT ] , 575 orientable, 573 universal, 570; covering group, 575 critical manifolds, 673 critical points and values, 28, 382-7 homotopically, 382, 387 index, 384 inessential, 383 nondegenerate, 383 (cross) section, 50, 4 1 6, 466 curl, 93 current 2-form J, 1 1 8 3-form .?i, 1 99 3-vector J, 1 1 9 4-vector J, 1 99 convective, 1 1 9 electric, as a chain, 644 curvature of a connection, 243 extrinsic, 3 1 8 forms () , 25 1 , 256, 43 1 ; and the Ad bundle, 489; of a surface, 257; () * on a frame bundle, 462; () * on a principal bundle, 48 1 Gauss, 207 geodesic, 235 intrinsic, 3 1 8 mean, 207 and parallel displacement, 259-6 1 of the Poincare metric, 258 principal, 207 Riemann sectional K (X /\ y ) , 3 1 3 Riemann tensor, 244 of a space curve, 1 9 1 of a surface, 207 of a surface of revolution, 258
686
curvature (continued) total, 2 1 5 transformation R ( X, Y ) , 244 vector, 1 92, 1 94 cycle absolute, 344 completeable, 387 group, 344 relative, 379 �\ 200 d ' Alembertian D, 293, 37 1 deformation retract, 406, 506 tensor, 82 theorem, 350 degree of a map, see Brouwer degree de Rham's theorem, 355-60 vector space �R P , 356 derivation, 1 34 derivative covariant, 235 exterior, 73 intrinsic, 235 normal, 364 determinant line bundle, 487 dictionary relating forms and vectors, 94 diffeomorphism, 27 differentiable, 20 differential exterior d, 73; covariant, 250 of a function, 40, of a map F* , 7, 27 differential form, see form differentiation of integrals, 1 3 8-43 Dirac adjoint or conjugate spinor, 532 algebra, 509 equation, 503 Lagrangian, 5 3 1 matrices, 5 1 0 monopole, 444; quantization, 445 operator, 5 1 1 , 5 1 4, 52 1 ; in curved space, 5 1 5-2 1 program, 502 representation p, 5 1 2 (4-component) spin or, 5 1 3 string, 1 62 Dirichlet's principle, 373 distance from a point to a hypersurface, 579
INDEX
distribution (of subspaces), 1 66 adiabatic, 1 83 horizontal, 263 integrable, 1 67 divergence, 93, 1 36, 304 exterior covariant, 545 of a form, 365 of a symmetric tensor, 300 theorem, 1 39 dual basis, 39 bundle, 4 1 7, 482 Hodge * , 362 space, 39 fj , 67 eigenvalue of a quadratic form, 63, 209 eight-fold way, 649 Einstein equations, 296, 3 1 6, 3 1 7 ; Wheeler's version, 318 geodesic assumption, 292, 297 tensor G, 3 1 5 electric field E, 1 1 9 I -form c.\, 1 20 2-form *{C" 1 2 1 and topology, 1 23, 378, 3 8 1 electromagnetic bundle, 44 1 connection, 440 field strength F 2, 1 97 Lagrangian, 308 stress-energy-momentum tensor, 308 vector potential I -form 1 99 electromagnetism and Maxwell's equations in curved spacetime, 366-7 existence and uniqueness, 378, 387 on projective space, 1 64 on the 3-sphere, 1 63 on the 3-torus, 1 22 embedded submanifold, 27 energy of deformation, 623-6 density, 3 1 6 hypersurface, 1 48; invariant form, I SO internal, 1 79 momentum vector, 1 95 momentum tensor, 295 of a path, 274 rest, 1 95 total, 1 48, 1 96 entropy, 1 83 empirical, 1 85
AI,
INDEX
equations of motion, 1 44 relativistic, 303 equilibrium equations, 630-2 euclidean metric in quantum fields, 55 1 Euler characteristic, 423, 426 equations of fluid flow, 1 44 integrability condition, 1 66 principle of least action, 28 1 exact form, 1 56 sequence, 598-600; homology, 604; homotopy, 600; short, 599 exp, 284 exponential map for a Lie group, 399, 403 extension theorem, 592 exterior algebra, 68 covariant differential V , 250, 430; of a form section of a vector bundle, 488 covariant divergence V*, 545 differential d , 73; coordinate expression, 76; spatial d, 1 4 1 form, 66; and vector analysis, 7 1 power operation, 588 product, 67; and determinants, 7 1 ; geometric meaning, 70 face, 335 Faraday's law, 1 2 1 Fermat's principle, 297 fermion, 658 fiber, 49, 4 1 5 bundle, 45 1 , 594 coordinate, 4 1 6 over p , 4 1 6 space, 593 field strength, 64 Flamm paraboloid, 32 1 flow generated by a vector field, 32, 33 by invariant fields, 408 by Lie bracket, 1 29 straightened, 35 fluid flow, 30, 1 43-5 magnetohydrodynamic, 1 45 foliation, 1 73 force classical, 1 95 Lorentz, 1 1 9 Minkowski, 1 95 form bi-invariant, 56 1 -3 Cartan, 562
687
Cauchy stress form, 6 1 8 closed, 1 56 exact, 1 56 exterior, 66 first fundamental, 202 harmonic, 370 heat I -form, 1 79 integration of, 95- 1 02; and pull-backs, 1 02 invariant, 395 Maurer-Cartan, 476 normal, 376 and pseudo-form, 1 22 p-form, 4 1 pseudo-, 86 pull-back, 77-82 second fundamental, 204, 309; and expansion of the universe, 3 1 8, 3 1 9 stress: Cauchy, 6 1 8 ; Piola-Kirchhoff, 622, 623 tangential, 376 of type Ad, 489, 490 vector bundle-valued, 429 vector-valued, dr and dS, 203, 248 volume, 86, 88 with values in a Lie algebra, 475 , 477 work I -form, 1 79 frame e, 243 change of, 253 coordinate, 243 orthonormal, 255 of sections, 4 1 7 frame bundle, 453 Frobenius chart, 1 67 theorem, 1 70 Frobenius-Schur relations, 664 functional derivative, 307 fundamental group JT I , 567-9, 578 theorem of algebra, 2 1 5 vector field, 455 v'g, 88 Galloway's theorem, 578 gauge bundle, 490 field, 255 , 536 invariance, 44 1 , 449, 533-6 particles: gluons, 540; mesons, 538; photons, 536 principle, 537 transformation, 255, 490; global, 535
688
INDEX
Gauss -Bonnet theorem, 2 1 5 , 323, 462 ; as an index theorem, 465 ; generalized, 465-8 curvature, 207 equations, 229, 3 1 0, 3 1 1 - 1 4; rel ativi stic meaning, 3 1 6- 1 8 formula for variation o f area, 225 law, 1 2 1 lemma, 286 l inking or looping integral , 2 1 8 normal map, 208, 2 1 5 , 260 theorem(l egreg illm, 23 1 , 3 1 7- 1 8 Gaussian coordinates, 284 Gell-Mann Gell-Mann matrices, 540, 652 Gel l-Mann/Okuba mass formula, 659 generalized momentum, 55 velocity, 50 general l inear group GI (n ) , 254, 39 1 general relativity, 29 1 -322 geodesic, 233, 27 1 -4 1 . Bernoulli 's theorem, 234 in a bi-i nvari ant metric, 563 circle, 287 closed, 28 1 , 284 completeness, 564 curvature Kg , 235, 239 equation, 235 null, 303 polar coordinates, 287 stability, 324, 326 submanifold, 3 1 0 ; total , 3 1 1 geodesy, 252 gluons, 540 gradient vector, 45 Grassmann algebra (see also exterior algebra) manifold, 459 Green's reciproci ty, 647 Green's theorem, 368 group JR, 2:, 2:2 , 336 boundary, 344 chain, 337 cycle, 344 de Rham, 356 exact sequence, 598 homology, 345 homomorphism, 337, 398
homotopy, 596 quotient, 345 :'ll, 200 Haar measure, 397, 5 4 1 Hadamard's lemma, 1 26 hairy sphere, 423 Hamilton, on composing rotations, 499 Hamilton's equations, 1 47 principle, 1 54, 275 ; in elasticity, 626-9 Hamiltonian, 1 47 flow, 1 48 operator, 439 relativistic, 1 96 vector field, 148 harmonic cochain, 64 1 harmonic field, 376 harmonic form , 370 i n a bi-invariant metric, 564 Hawking si ngularity theorem, 579 heat I -form, 1 79 helicity, 1 45 Helmholtz decomposition, 372 Hermitian adjoint t , 392 line bundle, 466 Hessian matri x, 383 Hi lbert action principle, 308 space inner product, 36 1 variational approach, 305-8 , 368 Hodge * operator, 362 codifferential d*, 364 decomposition, 372, 388 theorem, 37 1 theorem for normal forms, 3 8 1 theorem for tangential forms, 377 holomorphic, 1 5 8 holonomic constraint, 1 75 holonomy, 259 homeomorphism, 1 3 homogeneous space, 458 homologous, 345 homology group, 345-55 rel ative, 379; sequence, 604 homomorphism, 337, 398 algebra, 78 boundary, 338, 60 I induced, 337
INDEX
homotopic ally critical point, 382 homotopy, 59 1 and homology, 603 covering homotopy, 592 free homotopy class, 282, 283 sequence for a bundle, 600-3 homotopy groups J[k , 596-8 computation of, 605-8 and covering spaces, 605 of spheres, 597, 598 Hopf bundle, 473, 474 map and fibering, 606, 675 theorem, 427 Hopf-Rinow theorem, 564 horizontal distribution, 263-6, 48 1 Hurewicz theorem, 603 hypercharge, 654 hyperelastic, 626 hypersurface, 6 parallel, 286 1- and 2-sided, 84 immersion, 1 69, 1 7 3 i mplicit function theorem, 5 incidence matrix, 645 inclusion map, 79 index of a vector field (see also Kronecker index) of a section, 466 index theorem, 465 indicator, 3 1 5 infinitesimal generator, 399 instanton, 550 winding number, 556, 560 integrability condition, 1 66, 1 70, 1 74 integrable constraint, 1 75 distribution, 1 67 integral curve, 3 1 manifold, 1 66 integrating factor, 1 83 integration of forms, 96- 1 09 over manifolds, 1 04-9 of pseudoforms, 1 1 4 - 1 7 interaction, 534 interior product, 89 intersection number, 2 1 9 intrinsic, 234 derivative, 235
689
invariant form, 395 vector field, 395 volume form, 397 inverse function theorem, 29 image, 1 2 involution, 1 67 isometry, 230, 3 1 4 fixed set, 3 1 4 invariant, 23 1 isotopic spin, 648, 654 isotropic body, 66 1 isotropy subgroup, 457 1 , 432 Jacobi determinant, 5 equation of geodesic variation, 273 field, 1 29, 273, 326-9 identity, 403 metric, 28 1 principle of least action, 28 1 rule for change of variables in an integral, 101 variational equation, 1 28 Killing field, 528 equation, 529 kinetic term, 535 Kirchhoff's current law (KCL), 644 Kirchhoff's voltage law (KVL), 644 Klein bottle, 348 Klein-Gordon equation, 502 Kronecker delta, generalized oj , 67 index of a vector field, 2 1 6 Lagrange bracket { , }, 80, 1 00 deformation tensor, 82, 625 Lagrange's equations, 1 47 in a curved M 3 , 276 tensorial nature, 526 with electromagnetism, 439 Lagrangian, 54 Dirac, 5 3 1 electromagnetic, 308 for particle in an electromagnetic field, 436-9 significance in special relativity, 437
690
INDEX
Lambert's formula, 290 Lame moduli, 663 Laplace's formula for pressure in a bubble, 227 Laplacian V' 2 , 93, 305 and mean curvature, 305 Laplace operator on a cochain, 64 1 Laplace operator ,6. = dd* + d*d on forms, 368-72 on a I -form, 370 leaf of a foliation, 1 73 maximal, 1 73 Levi-Civita connection, 242 equation, 297 parallel displacement, 237 Lie algebra 3" 402 Ad invariant scalar product, 543 Lie bracket [ , ], 1 26, 402 Lie derivative S!x of a form, 1 32-8 of the metric tensor, 625 of the stress form, 634, 635 of a vector field, 1 25 Lie group, 39 1 -4 1 2 I -parameter subgroup, 398, 405-7, 564; on SI(2, JR) , 407 compact, 5 4 1 ; averaging over, 54 1 ; bi-invariant forms, 56 1 -7 connection and curvature of, 580 Lie subgroup and subalgebra, 4 1 0- 1 2 lifting paths, 277 in a bundle, 593 in a covering space, 574 lifting spheres, 605 light cone, 1 93 lightlike, 1 93 linear functional, 38 linking number, 2 1 9 Liouville's theorem, 148 local product, 49 trivialization, 4 1 7 Lorentz factor, 1 93 force, 1 1 9 ; covector, 1 20, 1 97 group, 504; and spinor representation of SI(2 , C), 509 metric, 1 92 transformation, 46, 1 98
magnetic field 8 , 1 1 9 I -form *fli, 1 2 1 2-form �I\, 1 20 and topology, 1 23, 387 magnetohydrodynamics, 1 45 manifold, 1 3, 1 9 closed, 1 20 complex, 2 1 integral, 1 66 mechanical, 1 80 orientable, 83 product, 1 5 pseudo-Riemannian, 45 Riemannian, 45 symplectic, 1 46 with boundary, 1 06 map canonical, 1 49 coordinate, 20 differentiable, 20 exponential, 284, 399 geographical, 230 inclusion, 79 of manifolds: critical points and values, 28; regular points and values, 28 projection, 4 1 5 matrix group, 394 Maurer-Cartan equations, 403, 477 form Q , 476 maximal atlas, 1 5 torus, 393 Maxwell's equations, 1 20-3 , 1 98 , 200, 536 on a curved space, 366--7 independence of, 200 on projective space, 1 64 on a 3-sphere, 1 63 on a torus, 1 22 Mayer-Lie system, 1 74 mean curvature, 207, 3 1 1 , 5 29 and divergence, 224 mesons, 538 Yukawa, 540 metric conformally related, 5 3 1 flat or locally euclidean, 263 Lorentz or Minkowski, 1 92 potentials, 293 pseudo-Riemannian, 45
INDEX
Riemannian, 45 spatial , 297 static, 292, 296 stationary, 29 1 tensor, 43 minimal submanifold, 3 1 1 , 528 surface, 227, 305 minimization of arc length, 286 Minkowski electromagnetic field tensor, 1 97 force, 1 95 metric and space, 46, 1 92 Mobius band, 1 8 mode normal , 65 zero, 465 momentum canonical, 439 classical, 1 94 density, 320, 3 22 4-vector, 1 94 generalized, 55 kinematical , 436 operator, 439 monopole bundle, 444, 473 Morse deformation, 47 equalities, 3 87, 428 index, 328, 384 inequal ities, 385, 386 lacunary principle, 388 lemma, 384 polynomial, 385 theory, 3 82-8 type number, 385, 604 multilinear, 5 8 Myers's theorem, 576-8 negative normal bundle, 673 neighborhood, 1 2 Noether's theorem, 527-9 Nomizu' s theorem, 530 normal bundle, 4 1 9, 6 1 6 coordinates, 287, 303 derivative, 364 map, 208 mode, 65 nucleon Heisenberg, 537 Yang-Mills, 538
obstruction cocycle, 609- 1 2 one parameter group, 3 1 open set, 1 1 , 1 2 orientability, 83 and curvature, 33 1 and homology, 349 and two-sidedness, 84 orientable bundle, 6 1 1 , 673 manifold, 83 transverse, 1 1 5 orientation, 82 of the boundary, 1 1 0 coherent, 34 1 transverse, 1 1 5 orthogonal group, O (n ) , 9, 392 5 0 (n ) , 9, 392 osculating plane, 1 9 1 paper folding, 3 1 5 parallel displacement, 237 independence of path, 260 parallelizable, 252 parameter, distinguished or affine, 272 parameterized subset, 97 partition of unity, 1 07 and Riemannian metrics, 1 09 passes peaks and pits, 427 path ordering, 555 Pauli algebra, 50 1 matrices, 493 period of a form, 357 periodic motion, 282 for double pendulum, 284 for rigid body, 3 3 1 Pfaffian, 1 67 phase, 448, 535 space, 5 5 ; extended, 1 5 1 physical components, 48, 630 Piola-Kirchhoff stress forms first, 622 second, 623 Poincare characteristic, 604 duality, 375 index theorem, 42 1 -8 lemma and converse, 1 60 metric, 239, 25 8 ; geodesics, 274, 530 I -form, 56; extended, 1 5 1
691
692 Poincare (continued) polynomial, 385 2-form, 80; extended, 1 5 1 , 437 Poisson bracket ( , ), 1 54 equation, 293, 37 1 potential of a closed form, 1 58, 1 60--4 global vector, 443, 448 monopole, 444 singularities, see Dirac string Poynting vector, 322 principal bundle, 454, 458, 48 1 directions, 207 , 3 1 0 normal, 1 9 1 normal curvatures, 207, 3 1 0 principle of least action, 28 1 probability amplitude, 447 projection, 49, 4 1 5 homomorphism, 605 projective space, 1 6, 85 homogeneous coordinates, 1 7 lR p n , 1 6 c p n , 22 proper time, 1 93 , 292 pseudoform, 86 integration of, 1 1 4 - 1 7 pseudo-Riemannian, 45 pull-back bundle, 622 of covariant tensors, 53, 77, 79 in elasticity, 8 1 , 622 and integration, 1 02 pure gauge, 553 quantization of a gauge field, 536 topological, 26 1 quark, 540 up, down, and strange flavored, 649 quasi-static, 1 79 quatemion, 502 quotient group, 345 radius of curvature, 1 92, 22 1 rate of deformation tensor, 632--4 regular points and values, 28 relative boundaries, cycles, and homology groups, 379-8 1 homology sequence, 604
INDEX
relativistic equations o f motion, 303 mass, 1 94 reparameterization , 1 0 1 representation, 48 1 adjoint, Ad, 486 dual, 482 irreducible, 662 of a group, 48 1 , 482 reducible, 65 1 tensor product, 482 residue of a form, 1 59 rest mass, 1 94 retraction, 2 1 7 Ricci curvature, 3 1 5 , 374, 577 identities 302 tensor Rij , 295 Riemann -Christoffel curvature tensor, 229 sectional curvature K (X /\ Y ) , 3 1 3- 1 4 sphere, 2 1 theorem, 266 Riemannian manifold and metric, 45; bi-invariant, 563 ; on a surface of revolution, 258 connection, 242 rigid body, 9, 33 1 rotation group S O (II ) , 392, 492 S ard's theorem, 29 scalar curvature R, 296 scalar product, 42 global, 36 1 of Hermitian matrices, 494 nondegenerate, 42 Schrodinger's equation, 439 in curved space, 442 with an electromagnetic field, 440, 443 Schur's lemma and corollary, 662, 663 Schwarz's formula, 228 Schwarzschild solution, 320-2 spatial metric, 298 section, 50, 4 1 6, 466 holomorphic, 467 p-form section of a vector bundle, 488 sectional curvature, 3 1 3 self adjoint, 205 , 3 1 7 self (anti) dual field, 549 Serret-Frenet formulas, 1 96, 43 1 Simon connection, 472
INDEX
simplex, 333 boundary, 335 face, 335 ordered, 335 orientation, 336 singular, 334 standard, 333 simplicial complex, 343 simply connected, 283, 329, 595 singularity of a vector field, 422 skeleton, 6 1 0 smooth, 7 soap bubbles and films, 226-8 space like, 1 93 space-time notation, 1 4 1 spatial slice, 3 1 6 special, 392 linear group, SI(n), 1 1 , 392 orthogonal group S O (n ) , 392 unitary group S U (n ) , 392 sphere lifting theorem, 605 spin structure, 5 1 5- 1 8 spinor adjoint, 532 bundle SM, 5 1 7 connection, 5 1 8-2 1 cospinor, 5 1 3 Dirac or 4-component, 5 1 3 group Spin(3), 497 "representation" of S O (3 ) , 497 "representation" of the Lorentz group, 509 2-component, 497 ; left- and right-handed, 5 1 3 stability, 324; subgroup, 457 Stiefel manifold, 459, 6 1 6 vector field, 426 Stokes's theorem, 1 1 1 - 1 4 generalized, 1 55 for pseudoforms, 1 1 7 stored energy of deformation, 623-6 strain energy, 660 stress-energy-momentum tensor 295 stress forms Cauchy, 6 1 8 ; symmetry, 620 first Piola-Kirchhoff, 622 second Piola-Kirchhoff, 623 stress tensor, 295, 6 1 7 structure constants, 402 in a bi-invariant metric, 566 structure group of a bundle, 433, 452 reduction, 433 S U (2) * U ( I ) , 657 S U (n ) , 392, 493-7
Tij'
693
subalgebra, 4 1 1 subgroup, 4 1 1 isotropy = little = stability, 457 submanifold, 26 embedded, 27 framed, 1 1 5 immersed, 1 69 of M n , 29 of lRn , 4, 8 1 - and 2-sided, 84 with transverse orientation, 1 1 5 submersion, 1 8 1 summation convention, 59 support, 1 07 symmetries, 527-3 1 symplectic form, 1 46 manifold, 1 46 Synge's formula, 325 theorem, 329 tangent bundle, 48 ; unit, 5 1 space, 7 , 25 vector, 23 Tel legen 's theorem, 646 tensor analysis, 298-303 Cauchy-Green, 82 contravariant, 59 covariant, 5 8 deformation, 8 2 , 625 metric, 5 8 mixed, 60; linear transformation, 6 1 product, 59, 66; representation, 482 rate of deformation, 632 transformation law, 62 two-point, 622 theorema egregium, 23 1 thermodynamics first law, 1 80 second law according to Lord Kelvin, 1 8 1 ; according to Caratheodory, 1 8 1 Thorn's theorem, 349 timelike, 1 93 topological invariants, 346 quantization, 468 topological space, 1 2 compact, 1 3 topology, 1 2 induced or subspace, 1 2
694
INDEX
torsion of a connection. 245 ; 2-form, 249 of a space curve, 1 96 torus. 1 6 maximal, 393 transformation group, 456 transition matrix cu v , 24, 254, 4 1 4 for the cotangent bundle, 4 1 7 for dual bundles, 4 1 7 for tangent bundle. 4 1 7 for tensor product bundle, 4 1 7 transitive, 456 translation (left and right), 393 transversal to a submanifold, 34 transverse orientati on, l i S triangulation, 346 tunnel i ng, 558 twisted product, 4 1 5 unitary group U (n ) , 392 universe static, 292 stationary, 29 1 vacuum state, 557, 558 tunneling, 558 vari ation of action, 1 54 external , 523 first, of arc length, 232; of area, 22 1 , 322 internal , 523 of a map, 1 53 of Ricci tensor, 306 second, of arc length, 324-32 variational derivative 0 , 307, 526 equation, 1 28 pri nciples of mechanics, 275-8 1 vector, 1 28, 1 53 , 272 vector analysis, 92, 1 36-8 bundle, 4 1 3- 1 9; -valued form. 488 contravari ant or tangent, 23 coordinate, 25 covariant = covector I -form, 4 1 as differential operator, 25 field, 25; flow ( I -parameter group) generated by, 32, 3 3 ; integral curve of, 3 1 ; along a submanifold, 269 gradient, 45 =
integral, 1 44, 308 invariant, 395 Killing. 528 product, 92, 94, 1 03 transformation law, 34 -valued form, 248 vari ational, 1 28, 1 53, 272 velocity 4-vector, 1 93 velocity field, 3 1 virtual displacement, 276 voltage as a cochain, 644 volume bundle, 488 form, 86, 88 invariant: i n mechanics, 1 48 ; on the energy hypersurface, I SO; on (he unit hyperboloid, 200; on a Lie group, 397, 54 1 ; on SI(2, lR) , 398 vorticity, 1 45 wedge product, see exterior product weight diagram, 655 Weingarten equations, 204 Weizenbock formulas, 370 Weyl's equation for neutrinos, 5 1 5 method of orthogonal proj ection, 647 principle of gauge invariance, 44 1 theorem on the fundamental group of a Lie group, 565, 5 8 1 Whitney embedding theorem, 23 winding number of a curve, 2 1 2 of a Yang-Mills instanton, 560; in terms of field strength, 5 85-7 of a Yang-Mills vacuum, 560 work I -form in thermodynamics, 1 79 world line, 1 93 wormhole, 446 Yang-Mills action, 544 analogy with electromagneti sm, 547, 548, 550 equations, 545 field strength, 539 i nstanton, 550; winding number, 560, 585 Yukawa-Kemmer, 658 2:2 , 3 36 zero modes, 465
